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自発的対称性の破れと 南部-Goldstone...
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自発的対称性の破れと 南部-Goldstone モード
日高義将(理研仁科センター)
2011年4月-2012年3月二期メンバー
前沢 畔柳飯田 村田
橋本 矢崎
木村 檜垣
Polchinski 8章まで読んだ
自発的対称性の破れと 南部-Goldstone モード
12031494 [hep-th] Phys Rev Lett 110 091601 (2013)
UV理論
IR理論SSB+真空 Chiral Lagrangian物質中 流体力学
Goldstone Salam Weinberg(rsquo62)南部-Goldstoneの定理
Lorentz対称性を持った真空大域的対称性の自発的破れ
破れた対称性(生成子)の数 = NGモードの数
分散関係
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
きっかけ2002~2004年あたりの理研の研究会橘さんがNielsen-Chadhaの論文を紹介()
Nucl Phys B 105 445 (1976)
Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
高密度QCD物質(K中間子凝縮相したカラー超伝導相)Miransky Shovkovy hep-ph0108178
SU(2)xU(1)rarrU(1)3つの破れた生成子2つのNGモード k k2
T1 T2 Y T3
非自明なNGモードの例強磁性体中のスピン波
スピン波(マグノン)
= plusmnv0k2
スピン対称性の破れ
hsz(n)i = m
cf hsz(n)i = (1)nm 反強磁性
2つのNGモード = plusmnv|k|
Bloch (1930)
自発的対称性の破れと NGモードの数分散の関係は よくわかっていなかった
Nielsen - Chadha(rsquo76)Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
Watanabe - Brauner (rsquo11)NBS NNG 1
2rankh[iQa Qb]i
Schafer Son Stephanov Toublan and Verbaarschot
NNG = NBS(rsquo01)
h[iQa Qb]i = 0
Nambu (rsquo04)h[iQa Qb]i 6= 0 (Qa Qb)
正準関係
NG定理の一般化
Watanabe Murayama (rsquo12)
YH (rsquo12)
NBS NNG =1
2rankh[iQa Qb]i
Ntype-A + 2Ntype-B = NBS
Ntype-B =1
2rankh[iQa Qb]i
最近の進展有効ラグランジアンの方法森の射影演算子法
連続対称性の自発的破れの定義
= |ih|真空
媒質中 =
exp((H microN))
tr exp((H microN))
自発的対称性の破れはある電荷Qaについて
となる局所場Φiが少なくとも一つは存在することで定義
h[iQai(x)]i tr [iQai(x)] 6= 0
もし電荷がwell-definedならば
自発的対称性の破れrArr電荷がill-defined
h[iQai(x)]i = tr[iQai(x)]
= tr[ iQa]i(x) = 0
[iQa ] = 0
cyclic property
F []
F []
縮退を伴う
場の場合連続対称性の自発的破れ
Ward-Takahashi IdentityF
ijh[iQaj ]i = 0
並進対称性が残っている場合弾性を伴う
a
スピンの場合
格子の場合
自由エネルギー
ギャップレスな励起が現れる= 南部-Goldstone(NG)モード
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
スピン波(マグノン)
格子振動(フォノン)
NGモードとは電荷密度は保存則により必ず遅い
電荷密度と弾性変数が正準共役cf Nambu (rsquo04)
対称性が自発的に破れると
Jmicroa = F
microa
相対論的な場合
媒質中 拡散方程式
例)tna(tx) = 2
i na(tx)
jia = ina
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
2011年4月-2012年3月二期メンバー
前沢 畔柳飯田 村田
橋本 矢崎
木村 檜垣
Polchinski 8章まで読んだ
自発的対称性の破れと 南部-Goldstone モード
12031494 [hep-th] Phys Rev Lett 110 091601 (2013)
UV理論
IR理論SSB+真空 Chiral Lagrangian物質中 流体力学
Goldstone Salam Weinberg(rsquo62)南部-Goldstoneの定理
Lorentz対称性を持った真空大域的対称性の自発的破れ
破れた対称性(生成子)の数 = NGモードの数
分散関係
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
きっかけ2002~2004年あたりの理研の研究会橘さんがNielsen-Chadhaの論文を紹介()
Nucl Phys B 105 445 (1976)
Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
高密度QCD物質(K中間子凝縮相したカラー超伝導相)Miransky Shovkovy hep-ph0108178
SU(2)xU(1)rarrU(1)3つの破れた生成子2つのNGモード k k2
T1 T2 Y T3
非自明なNGモードの例強磁性体中のスピン波
スピン波(マグノン)
= plusmnv0k2
スピン対称性の破れ
hsz(n)i = m
cf hsz(n)i = (1)nm 反強磁性
2つのNGモード = plusmnv|k|
Bloch (1930)
自発的対称性の破れと NGモードの数分散の関係は よくわかっていなかった
Nielsen - Chadha(rsquo76)Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
Watanabe - Brauner (rsquo11)NBS NNG 1
2rankh[iQa Qb]i
Schafer Son Stephanov Toublan and Verbaarschot
NNG = NBS(rsquo01)
h[iQa Qb]i = 0
Nambu (rsquo04)h[iQa Qb]i 6= 0 (Qa Qb)
正準関係
NG定理の一般化
Watanabe Murayama (rsquo12)
YH (rsquo12)
NBS NNG =1
2rankh[iQa Qb]i
Ntype-A + 2Ntype-B = NBS
Ntype-B =1
2rankh[iQa Qb]i
最近の進展有効ラグランジアンの方法森の射影演算子法
連続対称性の自発的破れの定義
= |ih|真空
媒質中 =
exp((H microN))
tr exp((H microN))
自発的対称性の破れはある電荷Qaについて
となる局所場Φiが少なくとも一つは存在することで定義
h[iQai(x)]i tr [iQai(x)] 6= 0
もし電荷がwell-definedならば
自発的対称性の破れrArr電荷がill-defined
h[iQai(x)]i = tr[iQai(x)]
= tr[ iQa]i(x) = 0
[iQa ] = 0
cyclic property
F []
F []
縮退を伴う
場の場合連続対称性の自発的破れ
Ward-Takahashi IdentityF
ijh[iQaj ]i = 0
並進対称性が残っている場合弾性を伴う
a
スピンの場合
格子の場合
自由エネルギー
ギャップレスな励起が現れる= 南部-Goldstone(NG)モード
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
スピン波(マグノン)
格子振動(フォノン)
NGモードとは電荷密度は保存則により必ず遅い
電荷密度と弾性変数が正準共役cf Nambu (rsquo04)
対称性が自発的に破れると
Jmicroa = F
microa
相対論的な場合
媒質中 拡散方程式
例)tna(tx) = 2
i na(tx)
jia = ina
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
Polchinski 8章まで読んだ
自発的対称性の破れと 南部-Goldstone モード
12031494 [hep-th] Phys Rev Lett 110 091601 (2013)
UV理論
IR理論SSB+真空 Chiral Lagrangian物質中 流体力学
Goldstone Salam Weinberg(rsquo62)南部-Goldstoneの定理
Lorentz対称性を持った真空大域的対称性の自発的破れ
破れた対称性(生成子)の数 = NGモードの数
分散関係
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
きっかけ2002~2004年あたりの理研の研究会橘さんがNielsen-Chadhaの論文を紹介()
Nucl Phys B 105 445 (1976)
Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
高密度QCD物質(K中間子凝縮相したカラー超伝導相)Miransky Shovkovy hep-ph0108178
SU(2)xU(1)rarrU(1)3つの破れた生成子2つのNGモード k k2
T1 T2 Y T3
非自明なNGモードの例強磁性体中のスピン波
スピン波(マグノン)
= plusmnv0k2
スピン対称性の破れ
hsz(n)i = m
cf hsz(n)i = (1)nm 反強磁性
2つのNGモード = plusmnv|k|
Bloch (1930)
自発的対称性の破れと NGモードの数分散の関係は よくわかっていなかった
Nielsen - Chadha(rsquo76)Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
Watanabe - Brauner (rsquo11)NBS NNG 1
2rankh[iQa Qb]i
Schafer Son Stephanov Toublan and Verbaarschot
NNG = NBS(rsquo01)
h[iQa Qb]i = 0
Nambu (rsquo04)h[iQa Qb]i 6= 0 (Qa Qb)
正準関係
NG定理の一般化
Watanabe Murayama (rsquo12)
YH (rsquo12)
NBS NNG =1
2rankh[iQa Qb]i
Ntype-A + 2Ntype-B = NBS
Ntype-B =1
2rankh[iQa Qb]i
最近の進展有効ラグランジアンの方法森の射影演算子法
連続対称性の自発的破れの定義
= |ih|真空
媒質中 =
exp((H microN))
tr exp((H microN))
自発的対称性の破れはある電荷Qaについて
となる局所場Φiが少なくとも一つは存在することで定義
h[iQai(x)]i tr [iQai(x)] 6= 0
もし電荷がwell-definedならば
自発的対称性の破れrArr電荷がill-defined
h[iQai(x)]i = tr[iQai(x)]
= tr[ iQa]i(x) = 0
[iQa ] = 0
cyclic property
F []
F []
縮退を伴う
場の場合連続対称性の自発的破れ
Ward-Takahashi IdentityF
ijh[iQaj ]i = 0
並進対称性が残っている場合弾性を伴う
a
スピンの場合
格子の場合
自由エネルギー
ギャップレスな励起が現れる= 南部-Goldstone(NG)モード
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
スピン波(マグノン)
格子振動(フォノン)
NGモードとは電荷密度は保存則により必ず遅い
電荷密度と弾性変数が正準共役cf Nambu (rsquo04)
対称性が自発的に破れると
Jmicroa = F
microa
相対論的な場合
媒質中 拡散方程式
例)tna(tx) = 2
i na(tx)
jia = ina
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
自発的対称性の破れと 南部-Goldstone モード
12031494 [hep-th] Phys Rev Lett 110 091601 (2013)
UV理論
IR理論SSB+真空 Chiral Lagrangian物質中 流体力学
Goldstone Salam Weinberg(rsquo62)南部-Goldstoneの定理
Lorentz対称性を持った真空大域的対称性の自発的破れ
破れた対称性(生成子)の数 = NGモードの数
分散関係
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
きっかけ2002~2004年あたりの理研の研究会橘さんがNielsen-Chadhaの論文を紹介()
Nucl Phys B 105 445 (1976)
Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
高密度QCD物質(K中間子凝縮相したカラー超伝導相)Miransky Shovkovy hep-ph0108178
SU(2)xU(1)rarrU(1)3つの破れた生成子2つのNGモード k k2
T1 T2 Y T3
非自明なNGモードの例強磁性体中のスピン波
スピン波(マグノン)
= plusmnv0k2
スピン対称性の破れ
hsz(n)i = m
cf hsz(n)i = (1)nm 反強磁性
2つのNGモード = plusmnv|k|
Bloch (1930)
自発的対称性の破れと NGモードの数分散の関係は よくわかっていなかった
Nielsen - Chadha(rsquo76)Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
Watanabe - Brauner (rsquo11)NBS NNG 1
2rankh[iQa Qb]i
Schafer Son Stephanov Toublan and Verbaarschot
NNG = NBS(rsquo01)
h[iQa Qb]i = 0
Nambu (rsquo04)h[iQa Qb]i 6= 0 (Qa Qb)
正準関係
NG定理の一般化
Watanabe Murayama (rsquo12)
YH (rsquo12)
NBS NNG =1
2rankh[iQa Qb]i
Ntype-A + 2Ntype-B = NBS
Ntype-B =1
2rankh[iQa Qb]i
最近の進展有効ラグランジアンの方法森の射影演算子法
連続対称性の自発的破れの定義
= |ih|真空
媒質中 =
exp((H microN))
tr exp((H microN))
自発的対称性の破れはある電荷Qaについて
となる局所場Φiが少なくとも一つは存在することで定義
h[iQai(x)]i tr [iQai(x)] 6= 0
もし電荷がwell-definedならば
自発的対称性の破れrArr電荷がill-defined
h[iQai(x)]i = tr[iQai(x)]
= tr[ iQa]i(x) = 0
[iQa ] = 0
cyclic property
F []
F []
縮退を伴う
場の場合連続対称性の自発的破れ
Ward-Takahashi IdentityF
ijh[iQaj ]i = 0
並進対称性が残っている場合弾性を伴う
a
スピンの場合
格子の場合
自由エネルギー
ギャップレスな励起が現れる= 南部-Goldstone(NG)モード
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
スピン波(マグノン)
格子振動(フォノン)
NGモードとは電荷密度は保存則により必ず遅い
電荷密度と弾性変数が正準共役cf Nambu (rsquo04)
対称性が自発的に破れると
Jmicroa = F
microa
相対論的な場合
媒質中 拡散方程式
例)tna(tx) = 2
i na(tx)
jia = ina
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
UV理論
IR理論SSB+真空 Chiral Lagrangian物質中 流体力学
Goldstone Salam Weinberg(rsquo62)南部-Goldstoneの定理
Lorentz対称性を持った真空大域的対称性の自発的破れ
破れた対称性(生成子)の数 = NGモードの数
分散関係
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
きっかけ2002~2004年あたりの理研の研究会橘さんがNielsen-Chadhaの論文を紹介()
Nucl Phys B 105 445 (1976)
Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
高密度QCD物質(K中間子凝縮相したカラー超伝導相)Miransky Shovkovy hep-ph0108178
SU(2)xU(1)rarrU(1)3つの破れた生成子2つのNGモード k k2
T1 T2 Y T3
非自明なNGモードの例強磁性体中のスピン波
スピン波(マグノン)
= plusmnv0k2
スピン対称性の破れ
hsz(n)i = m
cf hsz(n)i = (1)nm 反強磁性
2つのNGモード = plusmnv|k|
Bloch (1930)
自発的対称性の破れと NGモードの数分散の関係は よくわかっていなかった
Nielsen - Chadha(rsquo76)Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
Watanabe - Brauner (rsquo11)NBS NNG 1
2rankh[iQa Qb]i
Schafer Son Stephanov Toublan and Verbaarschot
NNG = NBS(rsquo01)
h[iQa Qb]i = 0
Nambu (rsquo04)h[iQa Qb]i 6= 0 (Qa Qb)
正準関係
NG定理の一般化
Watanabe Murayama (rsquo12)
YH (rsquo12)
NBS NNG =1
2rankh[iQa Qb]i
Ntype-A + 2Ntype-B = NBS
Ntype-B =1
2rankh[iQa Qb]i
最近の進展有効ラグランジアンの方法森の射影演算子法
連続対称性の自発的破れの定義
= |ih|真空
媒質中 =
exp((H microN))
tr exp((H microN))
自発的対称性の破れはある電荷Qaについて
となる局所場Φiが少なくとも一つは存在することで定義
h[iQai(x)]i tr [iQai(x)] 6= 0
もし電荷がwell-definedならば
自発的対称性の破れrArr電荷がill-defined
h[iQai(x)]i = tr[iQai(x)]
= tr[ iQa]i(x) = 0
[iQa ] = 0
cyclic property
F []
F []
縮退を伴う
場の場合連続対称性の自発的破れ
Ward-Takahashi IdentityF
ijh[iQaj ]i = 0
並進対称性が残っている場合弾性を伴う
a
スピンの場合
格子の場合
自由エネルギー
ギャップレスな励起が現れる= 南部-Goldstone(NG)モード
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
スピン波(マグノン)
格子振動(フォノン)
NGモードとは電荷密度は保存則により必ず遅い
電荷密度と弾性変数が正準共役cf Nambu (rsquo04)
対称性が自発的に破れると
Jmicroa = F
microa
相対論的な場合
媒質中 拡散方程式
例)tna(tx) = 2
i na(tx)
jia = ina
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
Goldstone Salam Weinberg(rsquo62)南部-Goldstoneの定理
Lorentz対称性を持った真空大域的対称性の自発的破れ
破れた対称性(生成子)の数 = NGモードの数
分散関係
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
きっかけ2002~2004年あたりの理研の研究会橘さんがNielsen-Chadhaの論文を紹介()
Nucl Phys B 105 445 (1976)
Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
高密度QCD物質(K中間子凝縮相したカラー超伝導相)Miransky Shovkovy hep-ph0108178
SU(2)xU(1)rarrU(1)3つの破れた生成子2つのNGモード k k2
T1 T2 Y T3
非自明なNGモードの例強磁性体中のスピン波
スピン波(マグノン)
= plusmnv0k2
スピン対称性の破れ
hsz(n)i = m
cf hsz(n)i = (1)nm 反強磁性
2つのNGモード = plusmnv|k|
Bloch (1930)
自発的対称性の破れと NGモードの数分散の関係は よくわかっていなかった
Nielsen - Chadha(rsquo76)Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
Watanabe - Brauner (rsquo11)NBS NNG 1
2rankh[iQa Qb]i
Schafer Son Stephanov Toublan and Verbaarschot
NNG = NBS(rsquo01)
h[iQa Qb]i = 0
Nambu (rsquo04)h[iQa Qb]i 6= 0 (Qa Qb)
正準関係
NG定理の一般化
Watanabe Murayama (rsquo12)
YH (rsquo12)
NBS NNG =1
2rankh[iQa Qb]i
Ntype-A + 2Ntype-B = NBS
Ntype-B =1
2rankh[iQa Qb]i
最近の進展有効ラグランジアンの方法森の射影演算子法
連続対称性の自発的破れの定義
= |ih|真空
媒質中 =
exp((H microN))
tr exp((H microN))
自発的対称性の破れはある電荷Qaについて
となる局所場Φiが少なくとも一つは存在することで定義
h[iQai(x)]i tr [iQai(x)] 6= 0
もし電荷がwell-definedならば
自発的対称性の破れrArr電荷がill-defined
h[iQai(x)]i = tr[iQai(x)]
= tr[ iQa]i(x) = 0
[iQa ] = 0
cyclic property
F []
F []
縮退を伴う
場の場合連続対称性の自発的破れ
Ward-Takahashi IdentityF
ijh[iQaj ]i = 0
並進対称性が残っている場合弾性を伴う
a
スピンの場合
格子の場合
自由エネルギー
ギャップレスな励起が現れる= 南部-Goldstone(NG)モード
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
スピン波(マグノン)
格子振動(フォノン)
NGモードとは電荷密度は保存則により必ず遅い
電荷密度と弾性変数が正準共役cf Nambu (rsquo04)
対称性が自発的に破れると
Jmicroa = F
microa
相対論的な場合
媒質中 拡散方程式
例)tna(tx) = 2
i na(tx)
jia = ina
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
きっかけ2002~2004年あたりの理研の研究会橘さんがNielsen-Chadhaの論文を紹介()
Nucl Phys B 105 445 (1976)
Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
高密度QCD物質(K中間子凝縮相したカラー超伝導相)Miransky Shovkovy hep-ph0108178
SU(2)xU(1)rarrU(1)3つの破れた生成子2つのNGモード k k2
T1 T2 Y T3
非自明なNGモードの例強磁性体中のスピン波
スピン波(マグノン)
= plusmnv0k2
スピン対称性の破れ
hsz(n)i = m
cf hsz(n)i = (1)nm 反強磁性
2つのNGモード = plusmnv|k|
Bloch (1930)
自発的対称性の破れと NGモードの数分散の関係は よくわかっていなかった
Nielsen - Chadha(rsquo76)Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
Watanabe - Brauner (rsquo11)NBS NNG 1
2rankh[iQa Qb]i
Schafer Son Stephanov Toublan and Verbaarschot
NNG = NBS(rsquo01)
h[iQa Qb]i = 0
Nambu (rsquo04)h[iQa Qb]i 6= 0 (Qa Qb)
正準関係
NG定理の一般化
Watanabe Murayama (rsquo12)
YH (rsquo12)
NBS NNG =1
2rankh[iQa Qb]i
Ntype-A + 2Ntype-B = NBS
Ntype-B =1
2rankh[iQa Qb]i
最近の進展有効ラグランジアンの方法森の射影演算子法
連続対称性の自発的破れの定義
= |ih|真空
媒質中 =
exp((H microN))
tr exp((H microN))
自発的対称性の破れはある電荷Qaについて
となる局所場Φiが少なくとも一つは存在することで定義
h[iQai(x)]i tr [iQai(x)] 6= 0
もし電荷がwell-definedならば
自発的対称性の破れrArr電荷がill-defined
h[iQai(x)]i = tr[iQai(x)]
= tr[ iQa]i(x) = 0
[iQa ] = 0
cyclic property
F []
F []
縮退を伴う
場の場合連続対称性の自発的破れ
Ward-Takahashi IdentityF
ijh[iQaj ]i = 0
並進対称性が残っている場合弾性を伴う
a
スピンの場合
格子の場合
自由エネルギー
ギャップレスな励起が現れる= 南部-Goldstone(NG)モード
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
スピン波(マグノン)
格子振動(フォノン)
NGモードとは電荷密度は保存則により必ず遅い
電荷密度と弾性変数が正準共役cf Nambu (rsquo04)
対称性が自発的に破れると
Jmicroa = F
microa
相対論的な場合
媒質中 拡散方程式
例)tna(tx) = 2
i na(tx)
jia = ina
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
非自明なNGモードの例強磁性体中のスピン波
スピン波(マグノン)
= plusmnv0k2
スピン対称性の破れ
hsz(n)i = m
cf hsz(n)i = (1)nm 反強磁性
2つのNGモード = plusmnv|k|
Bloch (1930)
自発的対称性の破れと NGモードの数分散の関係は よくわかっていなかった
Nielsen - Chadha(rsquo76)Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
Watanabe - Brauner (rsquo11)NBS NNG 1
2rankh[iQa Qb]i
Schafer Son Stephanov Toublan and Verbaarschot
NNG = NBS(rsquo01)
h[iQa Qb]i = 0
Nambu (rsquo04)h[iQa Qb]i 6= 0 (Qa Qb)
正準関係
NG定理の一般化
Watanabe Murayama (rsquo12)
YH (rsquo12)
NBS NNG =1
2rankh[iQa Qb]i
Ntype-A + 2Ntype-B = NBS
Ntype-B =1
2rankh[iQa Qb]i
最近の進展有効ラグランジアンの方法森の射影演算子法
連続対称性の自発的破れの定義
= |ih|真空
媒質中 =
exp((H microN))
tr exp((H microN))
自発的対称性の破れはある電荷Qaについて
となる局所場Φiが少なくとも一つは存在することで定義
h[iQai(x)]i tr [iQai(x)] 6= 0
もし電荷がwell-definedならば
自発的対称性の破れrArr電荷がill-defined
h[iQai(x)]i = tr[iQai(x)]
= tr[ iQa]i(x) = 0
[iQa ] = 0
cyclic property
F []
F []
縮退を伴う
場の場合連続対称性の自発的破れ
Ward-Takahashi IdentityF
ijh[iQaj ]i = 0
並進対称性が残っている場合弾性を伴う
a
スピンの場合
格子の場合
自由エネルギー
ギャップレスな励起が現れる= 南部-Goldstone(NG)モード
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
スピン波(マグノン)
格子振動(フォノン)
NGモードとは電荷密度は保存則により必ず遅い
電荷密度と弾性変数が正準共役cf Nambu (rsquo04)
対称性が自発的に破れると
Jmicroa = F
microa
相対論的な場合
媒質中 拡散方程式
例)tna(tx) = 2
i na(tx)
jia = ina
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
自発的対称性の破れと NGモードの数分散の関係は よくわかっていなかった
Nielsen - Chadha(rsquo76)Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
Watanabe - Brauner (rsquo11)NBS NNG 1
2rankh[iQa Qb]i
Schafer Son Stephanov Toublan and Verbaarschot
NNG = NBS(rsquo01)
h[iQa Qb]i = 0
Nambu (rsquo04)h[iQa Qb]i 6= 0 (Qa Qb)
正準関係
NG定理の一般化
Watanabe Murayama (rsquo12)
YH (rsquo12)
NBS NNG =1
2rankh[iQa Qb]i
Ntype-A + 2Ntype-B = NBS
Ntype-B =1
2rankh[iQa Qb]i
最近の進展有効ラグランジアンの方法森の射影演算子法
連続対称性の自発的破れの定義
= |ih|真空
媒質中 =
exp((H microN))
tr exp((H microN))
自発的対称性の破れはある電荷Qaについて
となる局所場Φiが少なくとも一つは存在することで定義
h[iQai(x)]i tr [iQai(x)] 6= 0
もし電荷がwell-definedならば
自発的対称性の破れrArr電荷がill-defined
h[iQai(x)]i = tr[iQai(x)]
= tr[ iQa]i(x) = 0
[iQa ] = 0
cyclic property
F []
F []
縮退を伴う
場の場合連続対称性の自発的破れ
Ward-Takahashi IdentityF
ijh[iQaj ]i = 0
並進対称性が残っている場合弾性を伴う
a
スピンの場合
格子の場合
自由エネルギー
ギャップレスな励起が現れる= 南部-Goldstone(NG)モード
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
スピン波(マグノン)
格子振動(フォノン)
NGモードとは電荷密度は保存則により必ず遅い
電荷密度と弾性変数が正準共役cf Nambu (rsquo04)
対称性が自発的に破れると
Jmicroa = F
microa
相対論的な場合
媒質中 拡散方程式
例)tna(tx) = 2
i na(tx)
jia = ina
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
Nielsen - Chadha(rsquo76)Ntype-I + 2Ntype-II NBS
Type-I Type-II k2n+1 k2n
Watanabe - Brauner (rsquo11)NBS NNG 1
2rankh[iQa Qb]i
Schafer Son Stephanov Toublan and Verbaarschot
NNG = NBS(rsquo01)
h[iQa Qb]i = 0
Nambu (rsquo04)h[iQa Qb]i 6= 0 (Qa Qb)
正準関係
NG定理の一般化
Watanabe Murayama (rsquo12)
YH (rsquo12)
NBS NNG =1
2rankh[iQa Qb]i
Ntype-A + 2Ntype-B = NBS
Ntype-B =1
2rankh[iQa Qb]i
最近の進展有効ラグランジアンの方法森の射影演算子法
連続対称性の自発的破れの定義
= |ih|真空
媒質中 =
exp((H microN))
tr exp((H microN))
自発的対称性の破れはある電荷Qaについて
となる局所場Φiが少なくとも一つは存在することで定義
h[iQai(x)]i tr [iQai(x)] 6= 0
もし電荷がwell-definedならば
自発的対称性の破れrArr電荷がill-defined
h[iQai(x)]i = tr[iQai(x)]
= tr[ iQa]i(x) = 0
[iQa ] = 0
cyclic property
F []
F []
縮退を伴う
場の場合連続対称性の自発的破れ
Ward-Takahashi IdentityF
ijh[iQaj ]i = 0
並進対称性が残っている場合弾性を伴う
a
スピンの場合
格子の場合
自由エネルギー
ギャップレスな励起が現れる= 南部-Goldstone(NG)モード
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
スピン波(マグノン)
格子振動(フォノン)
NGモードとは電荷密度は保存則により必ず遅い
電荷密度と弾性変数が正準共役cf Nambu (rsquo04)
対称性が自発的に破れると
Jmicroa = F
microa
相対論的な場合
媒質中 拡散方程式
例)tna(tx) = 2
i na(tx)
jia = ina
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
Watanabe Murayama (rsquo12)
YH (rsquo12)
NBS NNG =1
2rankh[iQa Qb]i
Ntype-A + 2Ntype-B = NBS
Ntype-B =1
2rankh[iQa Qb]i
最近の進展有効ラグランジアンの方法森の射影演算子法
連続対称性の自発的破れの定義
= |ih|真空
媒質中 =
exp((H microN))
tr exp((H microN))
自発的対称性の破れはある電荷Qaについて
となる局所場Φiが少なくとも一つは存在することで定義
h[iQai(x)]i tr [iQai(x)] 6= 0
もし電荷がwell-definedならば
自発的対称性の破れrArr電荷がill-defined
h[iQai(x)]i = tr[iQai(x)]
= tr[ iQa]i(x) = 0
[iQa ] = 0
cyclic property
F []
F []
縮退を伴う
場の場合連続対称性の自発的破れ
Ward-Takahashi IdentityF
ijh[iQaj ]i = 0
並進対称性が残っている場合弾性を伴う
a
スピンの場合
格子の場合
自由エネルギー
ギャップレスな励起が現れる= 南部-Goldstone(NG)モード
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
スピン波(マグノン)
格子振動(フォノン)
NGモードとは電荷密度は保存則により必ず遅い
電荷密度と弾性変数が正準共役cf Nambu (rsquo04)
対称性が自発的に破れると
Jmicroa = F
microa
相対論的な場合
媒質中 拡散方程式
例)tna(tx) = 2
i na(tx)
jia = ina
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
連続対称性の自発的破れの定義
= |ih|真空
媒質中 =
exp((H microN))
tr exp((H microN))
自発的対称性の破れはある電荷Qaについて
となる局所場Φiが少なくとも一つは存在することで定義
h[iQai(x)]i tr [iQai(x)] 6= 0
もし電荷がwell-definedならば
自発的対称性の破れrArr電荷がill-defined
h[iQai(x)]i = tr[iQai(x)]
= tr[ iQa]i(x) = 0
[iQa ] = 0
cyclic property
F []
F []
縮退を伴う
場の場合連続対称性の自発的破れ
Ward-Takahashi IdentityF
ijh[iQaj ]i = 0
並進対称性が残っている場合弾性を伴う
a
スピンの場合
格子の場合
自由エネルギー
ギャップレスな励起が現れる= 南部-Goldstone(NG)モード
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
スピン波(マグノン)
格子振動(フォノン)
NGモードとは電荷密度は保存則により必ず遅い
電荷密度と弾性変数が正準共役cf Nambu (rsquo04)
対称性が自発的に破れると
Jmicroa = F
microa
相対論的な場合
媒質中 拡散方程式
例)tna(tx) = 2
i na(tx)
jia = ina
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
F []
F []
縮退を伴う
場の場合連続対称性の自発的破れ
Ward-Takahashi IdentityF
ijh[iQaj ]i = 0
並進対称性が残っている場合弾性を伴う
a
スピンの場合
格子の場合
自由エネルギー
ギャップレスな励起が現れる= 南部-Goldstone(NG)モード
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
スピン波(マグノン)
格子振動(フォノン)
NGモードとは電荷密度は保存則により必ず遅い
電荷密度と弾性変数が正準共役cf Nambu (rsquo04)
対称性が自発的に破れると
Jmicroa = F
microa
相対論的な場合
媒質中 拡散方程式
例)tna(tx) = 2
i na(tx)
jia = ina
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
並進対称性が残っている場合弾性を伴う
a
スピンの場合
格子の場合
自由エネルギー
ギャップレスな励起が現れる= 南部-Goldstone(NG)モード
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
スピン波(マグノン)
格子振動(フォノン)
NGモードとは電荷密度は保存則により必ず遅い
電荷密度と弾性変数が正準共役cf Nambu (rsquo04)
対称性が自発的に破れると
Jmicroa = F
microa
相対論的な場合
媒質中 拡散方程式
例)tna(tx) = 2
i na(tx)
jia = ina
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
ギャップレスな励起が現れる= 南部-Goldstone(NG)モード
Nambu(rsquo60) Goldstone(61) Nambu Jona-Lasinio(rsquo61)
スピン波(マグノン)
格子振動(フォノン)
NGモードとは電荷密度は保存則により必ず遅い
電荷密度と弾性変数が正準共役cf Nambu (rsquo04)
対称性が自発的に破れると
Jmicroa = F
microa
相対論的な場合
媒質中 拡散方程式
例)tna(tx) = 2
i na(tx)
jia = ina
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
NGモードとは電荷密度は保存則により必ず遅い
電荷密度と弾性変数が正準共役cf Nambu (rsquo04)
対称性が自発的に破れると
Jmicroa = F
microa
相対論的な場合
媒質中 拡散方程式
例)tna(tx) = 2
i na(tx)
jia = ina
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
南部-Goldstoneの定理の仮定真空のLorentz対称性は破れていない
k2 = 0k = 0
通常スカラー場が凝縮NGモードはLorentzスカラー
(非相対論的時間と空間は対等でない)
(非相対論的4元ベクトルの凝縮もあり)電荷密度も凝縮可能
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
Type-A Type-B
2種類の励起
単振動 歳差運動
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
Type-ArarrType-B転移の古典模型
コマが付いた振り子
回転対称性は重力による陽な破れ
z軸の周りの回転は対称性がある
x y軸に沿った対称性は破れている
破れた対称性の数は2つ
もしコマが回っていなければふたつの独立な振動が存在もしコマが回っていたら一つの独立な(歳差)運動
Lx
Ly
= Lz
6= 0
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
最近の発展
Type-A Type-B単振動 歳差運動
内部対称性の自発的破れに伴うNGモードは 2つの振動のタイプに分類できる
Ntype-A = NBS 2Ntype-B
Watanabe Murayama (rsquo12) YH (rsquo12)
Ntype-B =1
2rankh[iQa Qb]i
NBS NNG =1
2rankh[iQa Qb]i
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
Type-A NG モード電荷密度と弾性変数が正準共役
Type-A (B)は Type-I (II) NG モードか
Type-A = Type-I
Type-B NG モード
Type-B = Type-II
電荷密度と電荷密度が正準共役
Hayata YH (14)
Hayata YH(14)
ik2
i|k|4
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
Type-A Type-B
2種類の励起
単振動 歳差運動 p
g g
重力
pk2 k2
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
Watanabe-Murayamaの方法Watanabe Murayama (rsquo12)
Leutwyler(rsquo94)
Lorentz対称性がない場合時間の1階微分の項も可能
L =1
2ab
ab +gab2
ab gab2
iai
b
+higher
作用が対称性の変換の元で不変
Watanabe Murayama (rsquo12)ab ih[Qa j
0b (x)]i
可能な有効Lagrangianを書き下す
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
電荷の分類
ldquoAlmost NG modesrdquoKapustin (rsquo12) Karasawa Gongyo (rsquo14)有効Lagrangian approach
交換関係の期待値
QAb QB
b Ai B
i
h[iQBa
Bi ]i 6= 0
破れた電荷 局所演算子
Type-A 0 0 0
Type-B 0 0or 0
QBa
QAa
はgappedになるBih[iQB
a Bi ]i 6= 0
YH (12rsquo) Hayata YH (rsquo14)
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
フェリ磁性
Gapped partnerの例
反強磁性スピンの大きさが同じなら
2つのType-A
フェリ磁性スピンの大きさが異なる
1つのgapped mode1つのtype-B
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
さらなる拡張について
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
様々な対称性の破れカイラル対称性
CC by-sa Aney
スピン対称性
U(1)対称性
CC by-sa Roger McLassus
並進対称性 並進対称性
CC by-sa Elijah van der Giessen
ガリレイ対称性
並進対称性
CC by-sa Didier Descouens
回転対称性
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
時空対称性の破れの例1格子振動並進(3つ)回転(3つ)ガリレイ(3つ)
回転とガリレイ変換に対応したギャップレスモードは
9個破れているしかし NGモードは並進の3つ
ない
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
例 弦
2つの破れ回転
NGモードは一つ
Low and Manohar (rsquo02)
並進
Px
Lzh(x)i秩序変数 y
x
string
Low - Manoharの議論
h(x)i
h[Px
]i = ix
hi 6= 0
h[Lz
]i = iyx
hi 6= 0
時空対称性の破れの例2
回転は並進を使って書けるので独立でない
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
非自明な例 液晶ネマティック相空間回転 O(3)rarrO(2)2つの破れた生成子2つの弾性変数
スメクティック-A 相回転の破れ O(3)rarrO(2)並進の破れ3つの破れた生成子1つの弾性変数残り回転は重たいモードに
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
Inverse Higgs mechanism
Inverse Higgs 機構
= eixmicroPmicroeiT
a
a(x)
Ivanov Ogievetsky (rsquo75) Low Manohar (rsquo02)
Maurer-Cartan 1形式crarr = i
1d = ie
iTaa
(d+ iPmicrodxmicro)eiT
aa
= Pmicrodxmicro + [T a
iPmicrodxmicro + d] + middot middot middot
= Pmicrodxmicro + T
a(microa + f
bamicro
b)dxmicro + middot middot middot
Volkov (rsquo73) Ogievetsky (rsquo74)
F []
平らな方向が破れた対称性の数に等しくない
Hayata YH (rsquo14)
Nicolis et al (rsquo13)Watanabe Brauner (rsquo14)Endlich Nicolis Penco (rsquo13) YH Noumi Shu (rsquo14)
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
分散関係例)液晶 (Type-A)
回転 O(3)rarrO(2)ネマティック相
分散関係実部と虚部が同じオーダー(減衰振動)
の時 過減衰
Li(x) = ijkxjT
0k(x) i = 1 2
a = 0
例) 表面張力波 (Type-B)
Hosino Nakano(rsquo82)
k32
= ak2 + ibk2
NBS = NEV = 2
1
Vh[Pz N ]i 6= 0
Effective Lagrangian Watanabe Murayama (rsquo14)cf Takeuchi Kasamatsu (13)
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
分散関係
液晶(smectic-A相)
1次元的な秩序
$
amp
amp
amp
(
()
amp
amp)
ampamp
ampamp)
+-0-12345354167138191465lt=12303gt1A11
BC840-16lt=lt203D1A1
B1$)16lt=lt203D1A1
5
Solution to dense QCD in 1+1 dimensionsBringoltz 09014035 lsquot Hooft model with massive quarks
Works in Coulomb gauge in canonical ensemble fixed baryon number
Solves numerically equations of motion under constaint of nonzero baryon
Finds chiral density wave
NB for massive quarks should have massless excitations but with energy
~1Nc
24
= plusmnq
ak2z + bk4
= plusmn
sk2(ak
2z + bk4)
k2 + k2z
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
トポロジカルソリトン
並進と内部対称性
並進と並進
Kobayashi Nitta (14)
Watanabe Murayama (14)
例) domain wall in nonrelativistic massive CP1 model
例) 2+1D skyrmion Kelvin wave
[Px
Py
] N
z並進
topological number y並進
[Pz Q] Ntopological number
x並進
U(1)電荷
(several tens of nanometres) can be regarded as a magnetically 2Dsystem in which the direction of q is confined within the planebecause the sample thickness is less than the helical wavelengththerefore various features should appear that are missing in bulksamples In the context of the skyrmion the thin film has the advant-age that the conical state is not stabilized when the magnetic field isperpendicular to the plane23 Therefore it is expected that the SkX canbe stabilized much more easily and even at T 5 0 in a thin film ofhelical magnet
In this Letter we report the real-space observation of the forma-tion of the SkX in a thin film of B20-type Fe05Co05Si the thickness ofwhich is less than the helical wavelength using Lorentz TEM28 with ahigh spatial resolution The quantitative evaluation of the magneticcomponents is achieved by combining the Lorentz TEM observationwith a magnetic transport-of-intensity equation (TIE) calculation(Supplementary Information)
We first discuss the two prototypical topological spin texturesobserved for the (001) thin film of Fe05Co05Si The Monte Carlosimulation (Supplementary Information) for the discretized versionof the Hamiltonian in equation (1) predicts that the proper screw(Fig 1a) changes to the 2D skyrmion lattice (Fig 1b) when a perpen-dicular external magnetic field is applied at low temperature and whenthe thickness of the thin film is reduced to close to or less than thehelical wavelength The Lorentz TEM observation of the zero-fieldstate below the magnetic transition temperature (40 K) clearlyreveals the stripy pattern (Fig 1d) of the lateral component of themagnetization with a period of 90 nm as previously reported18 thisindicates the proper-screw spin propagating in the [100] or [010]direction When a magnetic field (50 mT) was applied normal to theplate a 2D skyrmion lattice like that predicted by the simulation(Fig 1b) was observed as a real-space image (Fig 1e) by means ofLorentz TEM The hexagonal lattice is a periodic array of swirling spintextures (a magnified view is shown in Fig 1f) and the lattice spacing isof the same order as the stripe period 90 nm Each skyrmion has theDzyaloshinskiindashMoriya interaction energy gain and the regionsbetween them have the magnetic field energy gain Therefore theclosest-packed hexagonal lattice of the skyrmion has both energygains and forms at a magnetic field strength intermediate betweentwo critical values each of which is of order a2J in units of energy We
note that the anticlockwise rotating spins in each spin structure reflectthe sign of the DzyaloshinskiindashMoriya interaction of this helical mag-net Although Lorentz TEM cannot specify the direction of the mag-netization normal to the plate the spins in the background (where theblack colouring indicates zero lateral component) should pointupwards and the spins in the black cores of the lsquoparticlesrsquo should pointdownwards this is inferred from comparison with the simulation ofthe skyrmion and is also in accord with there being a larger upwardcomponent along the direction of the magnetic field The situation issimilar to the magnetic flux in a superconductor29 in which the spinsare parallel to the magnetic field in the core of each vortex
Keeping this transformation between the two distinct spin textures(helical and skyrmion) in mind let us go into detail about their fieldand temperature dependences First we consider the isothermal vari-ation of the spin texture as the magnetic field applied normal to the(001) film is increased in intensity The magnetic domain configura-tion at zero field is shown in Fig 2a In analogy to Bragg reflectionsobserved in neutron scattering22 two peaks were found in the cor-responding fast Fourier transform (FFT) pattern (Fig 2e) confirm-ing that the helical axis is along the [100] direction In the real-spaceimage however knife-edge dislocations (such as that marked by anarrowhead in Fig 2a) are often seen in the helical spin state aspointed out in ref 18 When a weak external magnetic field of20 mT was applied normal to the thin film the hexagonally arrangedskyrmions (marked by a hexagon in Fig 2b) started to appear as thespin stripes began to fragment The coexistence of the stripe domainand skyrmions is also seen in the corresponding FFT pattern (Fig 2f)the two main peaks rotate slightly away from the [100] axis and twoother broad peaks and a weak halo appear With further increase ofthe magnetic field to 50 mT (Fig 2c) stripe domains were completelyreplaced by hexagonally ordered skyrmions Such a 2D skyrmionlattice structure develops over the whole region of the (001) sampleexcept for the areas containing magnetic defects (SupplementaryInformation) A lattice dislocation was also observed in the SkX asindicated by a white arrowhead in Fig 2c The corresponding FFT(Fig 2g) shows the six peaks associated with the hexagonal SkXstructure The SkX structure changes to a ferromagnetic structureat a higher magnetic field for example 80 mT (Fig 2d h) renderingno magnetic contrast in the lateral component
d e f
90 nm 90 nm 30 nm
[010] [100]
a b c
Figure 1 | Topological spin textures in the helical magnet Fe05Co05Sia b Helical (a) and skyrmion (b) structures predicted by Monte Carlosimulation c Schematic of the spin configuration in a skyrmion dndashf Theexperimentally observed real-space images of the spin texture representedby the lateral magnetization distribution as obtained by TIE analysis of the
Lorentz TEM data helical structure at zero magnetic field (d) the skyrmioncrystal (SkX) structure for a weak magnetic field (50 mT) applied normal tothe thin plate (e) and a magnified view of e (f) The colour map and whitearrows represent the magnetization direction at each point
LETTERS NATURE | Vol 465 | 17 June 2010
902Macmillan Publishers Limited All rights reservedcopy2010
Yu et al Nature 465 901 (2010)
Kobayashi Nitta (rsquo12)
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
並進と内部対称性の破れ
Magnon RipplonType-A Type-A
Ripplon-MagnonType-B
[QPz] = 0 [QPz] 6= 0
Kobayashi Nitta 14026826
domain wall解の周りのNGモードCP1模型
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
NG mode in Active matter
(フォッカープランク)方程式に 対称性があるが保存しないの自発的破れ
CC BY-SA 20
Minami YH (rsquo15)拡散モードが現れる = ik2(保存系の場合伝搬モード)
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
まとめ
内部対称性
Ntype-B =1
2rankh[iQa Qb]i
Ntype-A = NBS Ntype-B
時空対称性Type-A
Type-B = ak ibk2
= a0k2 ib0k4
分散関係
一般ルールは
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
Super symmetry in condensed matterType-B NG fermion Satow Blaizot YH (rsquo15)
空気中を伝わる音波は 自発的
光(フォトン)はNGモードして解釈可能
Fermi流体のゼロ音波はトポロジカル絶縁体のエッジモードは
Ferrari Picasso (rsquo71) Hata (rsquo82) Kugo Terao Uehara (rsquo85) Hayata YH (rsquo14)
SSB of Generalized Global symmetryGaiotto Kapustin Seiberg Willett (rsquo14)
Generalized Global symmetryとの関係は
Effective theory for spacetime symmetry breakingYH Noumi Shu (rsquo14)
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
音波と光の類似性音波 H =
1
2ee+
1
2hpipi
e(x) pi(x)P = hi(x y)
光 H =1
2E2
i +1
2B2
i
[Bj(x) Ei(y)] = iikjk(x y)
Bi = ijkjAk
pi = hi
これから
量子非平衡系
これから
量子非平衡系