Symposium on Novel Quantum States in Condensed …nqs2017.ws/Slides/Sympo/10th/...Symposium on Novel...
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Symposium on Novel Quantum States in Condensed Matter 2017 (NQS2017)
8-10 November 2017 YITP, Kyoto University
Half-integer thermal Hall conductance in a Kitaev spin liquid
− Evidence for chiral Majorana edge current −
Yuichi KASAHARA Department of Physics, Kyoto University
hκxy
2D/T [( π/6)(kB 2 /
)]
hκxy
2D/T [(π/6)(kB 2 /
)]h
κxy
2D/T [(π/6)(kB 2 /
)]
hκxy
2D/T [( π/6)(kB 2 /
)]
1.5
1.0
0.5
0
-0.5
κ xy/T
(10
-3 W
/K2 m
)
5.25.04.84.64.4µ0H⊥ (T)
0 1/2
9.08.58.07.5µ0H|| (T)
4.3 K
1.5
1.0
0.5
0.0
-0.5
κ xy/T
(10
-3 W
/K2 m
)
5.25.04.84.64.4µ0H⊥ (T)
0 1/2
9.08.58.07.5µ0H|| (T)
4.9 K2.0
1.5
1.0
0.5
0.0
-0.5
κ xy/T
(10
-3 W
/K2 m
)
6.05.55.04.54.0µ0H⊥ (T)
0 1/2 1
10.09.08.07.0µ0H|| (T)
5.6 K
8.2 K
1.5
1.0
0.5
0
-0.5
κ xy/T
(10
-3 W
/K2 m
)
5.25.04.84.64.4µ0H⊥ (T)
0 1/2
9.08.58.07.5µ0H|| (T)
3.9 K
a b
c d
cold
hot
Majorana fermion
Z2 flux
Chiral Majorana edge currentHalf-integer thermal Hall conductance
Collaborators
Kaori Sugii, Masaaki Shimozawa, Minoru Yamashita Institute for Solid State Physics, The University of Tokyo
Taka Shibauchi Department of Advanced Materials Science, The University of Tokyo
Takafumi Onishi, Yuji Matsuda Department of Physics, Kyoto University
Nobuyuki Kurita, Hidekazu Tanaka Department of Physics, Tokyo Institute of Technology
Joji Nasu Department of Physics, Tokyo Institute of Technology
Yukitoshi Motome Department of Applied Physics, The University of Tokyo
Matsuda
Shibauchi
Tanaka
Nasu
Yamashita
Shimozawa
Sugii
Motome
Outline
1. Introduction: Kitaev quantum spin liquid
2. A candidate of Kitaev magnet α-RuCl3
3. Thermal Hall effect in perpendicular fields
4. Thermal Hall effect in tilted fields Observation of half-integer thermal Hall conductance
5. Summary
Y. Kasahara et al., arXiv:1709.10286 (2017).
IntroductionQuantum spin liquid (QSL)
Quantum fluctuations melt the long-range magnetic order even at T = 0.
1D: S = 1/2 XXZ chain
2D & 3D: Geometrically frustrated magnets2D trianglar 2D kagome 3D pyrochlore
Exotic physical properties in QSLsTopological phases Gauge fluctuations Fractionalized excitations
Spin liquid are states which do not break any simple symmetry: Neither spin-rotational symmetry nor lattice translational symmetry.
Platforms of QSL
The ground state with massive entanglement of local spins.
4
H = �Jx
X
<ij>
x
Sx
i
Sx
j
� Jy
X
<ij>
y
Sy
i
Sy
j
� Jz
X
<ij>
z
Sz
i
Sz
j
Kitaev model
Kitaev Interaction Bond-dependent Ising-like interaction
Honeycomb lattice (2D)A. Kitaev, Ann. Phys. 321, 2 (2006).
Hyper-honeycomb lattice (3D)S. Mandal & N. Surendran, PRB 79, 024426 (2009).
Exchange frustration
�Jx
Sx
i
Sx
j
�JzSzi S
zj
�JySyi S
yj �JyS
yi S
yj
�Jx
Sx
i
Sx
j
�JzSzi S
zj
S = 1/2 spins on tri-coordinate lattices A. Kitaev, Ann. Phys. 321, 2 (2006).
HK = �Jx
X
<ij>
x
Sx
i
Sx
j
� Jy
X
<ij>
y
Sy
i
Sy
j
� Jz
X
<ij>
z
Sz
i
Sz
j
Kitaev model
Fractionalization of quantum spins
Wp = +1
HK =iJ
x
4
X
<ij>
x
ci
cj
� iJy
4
X
<ij>
y
ci
cj
� iJz
4
X
<ij>
z
⌘r
ci
cj ⌘r = ic̄ic̄j
Spin 1/2Itinerant Majorana fermion with Dirac cone dispersion
Localized Majorana fermion
Z2 fluxes
Si
ci
c̄iWp = ⌘r⌘r0
Two types of QSLs
Gappless QSL: Majorana metalGapped QSL: Toric code
A. Kitaev, Ann. Phys. 321, 2 (2006).Wp = �1
Z2 flux Itinerant Majorana fermion
6
Spin fractionalization occurs below ~JK/kB. (proximate spin liquid state)
Jordan-Wigner transformation
Majorana representation
Free Majorana fermions on a honeycomb lattice
Candidate materials
G. Jackeli & G. Khaliullin, PRL 102, 017205 (2006).
Spin-orbit assisted Mott insulator with j = 1/2
90° bond formed by edge-shared octahedra
7
4d5 or 5d5
Candidate materials
β-Li2IrO3
2D honeycomb lattice 3D hyper-honeycomb lattice
Y. Singh & P. Gegenwart, PRB 82, 064412 (2010).
T. Takayama et al., PRL 114, 077202 (2015).
K. W. Plumb et al., PRB 90, 041112 (2014).
α-RuCl3
Na2IrO3
8
J = �1.7meV
K = �6.7meV
� = +6.6meV
Layered honeycomb magnet α-RuCl3
•AFM order with zigzag spin structure at TN ~ 7.5 K
•Transition at 14 K appears due to stacking faults.
J. A. Sears et al., PRB 91, 144420 (2015).
S. M. Winter et al., PRB 93, 214431 (2016).
Presence of non-Kitaev interaction
Dominant Kitaev term JK/kB ~ 100 K
9
H =X
<ij>
[J ~Si · ~Sj + JKS�i S
�j + �(S↵
i S�j + S�
i S↵j )]
Heisenberg Kitaev off-diagonal exchange
Kx = −6.7 meV, Ky = −6.7 meV, Kz = −5.0 meV
Possible signatures of Kitaev QSL in α-RuCl3Raman scattering
Fermionic excitations
L. J. Sandilands et al., PRL 114, 147201 (2015).
J. Nasu et al., Nat. Phys. 12, 912 (2016).
Broad magnetic continuum at high energy
Inelastic neutron scattering
Broad magnetic continuum appears below ~ JK/kB
S.-H. Do et al., Nat. Phys. http://doi.org/10.1038/nphys4298. A. Banerjee et al., Nat. Mater. 15, 733 (2016). A. Banerjee et al., Science 356, 1055 (2017).
Experiment
Theory
Possible signature of spin fractionalization
10More direct measurements are required.
What gives direct signature of Majorana fermions?
Topological system characterized by Chern insulator under H
Chiral edge current of Majorana fermions
H = 0 H ≠ 0
H = HK +He↵h
He↵h
= �h̃X
(ijk)
Sx
i
Sy
j
Sz
k
h̃ = �h3 ⇠ h3
�2f
�f ⇠ 0.06JK
HK = �Jx
X
<ij>
x
Sx
i
Sx
j
� Jy
X
<ij>
y
Sy
i
Sy
j
� Jz
X
<ij>
z
Sz
i
Sz
j
Effect of magnetic field
Massless Dirac cone
Massive Dirac cone
(h || [111]) A. Kitaev, Ann. Phys. 321, 2 (2006).
Flux gap
11
Non Abelian phase
�2Dxy
= ⌫e2
h
What gives direct signature of Majorana fermions?Kitaev QSLInteger QHE
q =⌫
2
Half-integer thermal Hall conductance in a Kitaev QSL
ν : Chern number 2Dxy
T= q
⇡
6
k2B
~2Dxy
T= ⌫
⇡
6
k2B
~
q : Central charge
cold
hot cold
hot
electronMajorana fermion
B B
Z2 flux
Chiral edge current of charge neutral Majorana fermions
= # of chiral edge modes
Chiral edge current of electrons
2Dxy
T=
1
2
✓⇡
6
k2B~
◆q = v in IQHE
12
Thermal Hall effect in insulating magnets✓
q0
◆=
✓xx
xy
�xy
xy
◆✓�rT
x
�rTy
◆
Ferromagnetically ordered state
Paramagnetic state
Spin liquid state
D. Watanabe, PNAS 113, 8653 (2016).Cu3V2O7(OH)2・2H2O
Lu2V2O7 Ho2V2O7, In2Mn2O7, BiMnO3 Cu(1-3,bdc)
Magnon Hall effect arising from Berry phase
Tb2Ti2O7
Spinon Hall effect in QSL state with spinon Fermi surface
M. Hirschberger et al., Science 348, 106 (2015).
Y. Onose et al, Science 329, 297 (2010). Ideue et al., PRB 85, 134411 (2012). M. Hirschberger et al., PRL 115, 106603 (2015).
spinon
xx
= 2⇡2
3
⇣"F
~ ⌧⌘ k2
B
T
h
1
dspinon
xy
= spinon
xx
(!c
⌧)
H. Katsura et al., PRL 104, 066403 (2010).
R. Matsumoto et al., PRB 89, 054420 (2014).
Lu2V2O7
Thermal transport measurements in α-RuCl3
14
• Clear anomaly at TN ~ 7.5 K • No discernible anomaly at
~ 14 K due to stacking faults
High quality single crystal
✓q0
◆=
✓xx
xy
�xy
xy
◆✓�rT
x
�rTy
◆
Magnetic susceptibility Specific heat28
26
24
22
20
18
χ (1
0-3 e
mu/
mol
)
20151050
T (K)
Sample 1
Sample 2
Thermal Hall effect
ex.) Spin liquid state: Kagome volborthite Cu3V2O7(OH)2・2H2O
Magnetically ordered state: Kagome Cu-(1-3,bdc) Pyrochlore Lu2V2O7
0 T
spinon
magnon
cf.) Phonon thermal Hall effect
κxy/T ~ 10-5 W/K2m
κxy/T ~ 10-6 W/K2m
κxy/T ~ 10-5 - 10-4 W/K2m
α-RuCl3 single crystals
Longitudinal thermal conductivity κxx
15
xx
= spxx
+ phxx
spin phonon
35
30
25
20
15
10
5
0
κ xx
(W/K
m)
706050403020100
T (K)
Sample 10 T12 T
Sample 20 T12 T
TN
12
8
4
0151050
TN
-0.08
-0.04
0.00
Δκ xx(H
)/κxx
(0)
1612840
µ0H (T)
50 K
20 K
12 K
35 K 27 K
Sample 2
However, it is difficult to separate spin & phonon contributions.
• Clear anomaly in κxx at TN
• Suppression of κxx by magnetic field κxxph is usually enhanced due to suppression of spin-phonon scattering by spin polarization.
Thermal Hall effectThermal transport is governed by spin excitations.
Thermal Hall conductivity κxy
Finite κxy ~ 10-2 W/Km at T < JK/kB
-2
0
2
κ xy
(10-3
W/K
m)
-10 0 10µ0H (T)
7 K
12 KT > TN
T < TN
Distinct H-dependence below and above TN • Sign change below TN
• Upward curvature above TN but downward below TNe.g.) κxy < 10-3 W/Km
in volborthite (spin liquid) Tb2Ti2O7 (paramagnet)
16
Thermal Hall effect below and above TN is different in origin.
8
6
4
2
0
-2
-4
κ xy/T
(10
-4 W
/K2 m
)
806040200
T (K)
0.2
0.1
0
-0.1
(κxy
2D/T)/(kB
2/h)
π/12
Sample 1 (6 T)Sample 1 (12 T)Sample 2 (15 T)
TN
0.2
0.1
0
-0.1
(κxy
2D
/T)/(k B
2 /h
)1.00.50.0
T/(JK/kB)
h*/JK = 0.030.06
π/12
Thermal Hall conductivity κxy
17
• Spin liquid with spinon Fermi surface
• Enhancement of κxy with positive sign below JK/kB ~ 80 K
• Broad peak at ~ 20 K
A. V. Inyushkin & . N. Taldenkov, JETP Lett. 86, 379 (2007).
• Phonons
• Magnons
Small DM interaction D/kB ~ 5 K << J/kB ~ 80 K
κxy/T ~ 10-6 W/K2m
Finite κxy/T usually appears in the ordered state.
• Exotic quasiparticle excitations inherent to the spin-liquid state of α-RuCl3.
D. Watanabe, PNAS 113, 8653 (2006).
In volborthite, Hall signal is negative.
κxy/T ~ 10-5 W/K2m
S. M. Winter et al., PRB 93, 214431 (2016).
Different T-dependence
Comparison with numerical calculations
18Y. Kasahara et al., arXiv:1709.10286 (2017).
T-dependence is consistent with numerical calculations for the 2D pure Kitaev model.
J. Nasu, J. Yoshitake & Y. Motome, PRL 119, 127204 (2017).
• Enhancement of κxy with positive sign below T < JK/kB
• Broad peak at T ~ 0.1JK/kB
• κxy/T reaches close to half of the quantization value.
Experiments Calculations
Possible signature of Majorana fermion excitations
2Dxy
= xy
d
interlayer distance d = 5.72 Å
thermal Hall conductance per 2D honeycomb layer
J. Nasu, J. Yoshitake & Y. Motome, PRL 119, 127204 (2017).
2Dxy
= xy
d
interlayer distance d = 5.72 Å
thermal Hall conductance per 2D honeycomb layer
Experiments Calculations
Comparison with numerical calculations
19Y. Kasahara et al., arXiv:1709.10286 (2017).However, quantization of κxy2D/T is not attained due to the magnetic order.
T-dependence is consistent with numerical calculations for the 2D pure Kitaev model.• Enhancement of κxy with positive sign below T < JK/kB
• Broad peak at T ~ 0.1JK/kB
• κxy/T reaches close to half of the quantization value.
Possible signature of Majorana fermion excitations
Suppression of AFM order by in-plane fields
A. Banerjee et al., arXiv:1706.0703 (2017).
Hc|| ~ 7-8 TM. Majumder et al., PRB 91, 180401(R) (2015).
AFM order is little influenced by out-of-plane fields, but it is easily suppressed by in-plane fields.
Low-temperature properties are masked by the magnetic order.
Key questions: Is the magnetic order suppressed by tuning parameters? Whether Kitaev QSL survives when suppressing the order?
20
Suppression of AFM order by in-plane fields
T�11 / 1
Texp
✓�n�
T
◆NMR: Unusual spin gap
N. Jansa et al., arXiv:1706.08455 (2017).
NMR
A. Banerjee et al., arXiv:1706.0703 (2017).
Hc|| ~ 7-8 T
Low-temperature properties are masked by the magnetic order.
21
Key questions: Is the magnetic order suppressed by tuning parameters? Whether Kitaev QSL survives when suppressing the order?
NMR: Unusual spin gap
Neutron scattering: Magnetic continuum at high energy above Hc||
Suppression of AFM order by in-plane fields
A. Banerjee et al., arXiv:1706.0703 (2017).
Inelastic neutron scatteringSpin wave
Continuum Continuum
T�11 / 1
Texp
✓�n�
T
◆
Low-temperature properties are masked by the magnetic order.Key question: Kitaev QSL survives when suppressing the magnetic order?
Kitaev QSL emerges under in-plane magnetic fields.
22
What gives direct signature of Majorana fermions?
Phase transition is tuned by H|| = Hsinθ. Thermal Hall response is determined by H⊥ = Hcosθ.
Measurements of thermal Hall effect in tilted fields
Thermal Hall effect in a Kitaev QSL state
Low-temperature properties are masked by the magnetic order.
Key questions: The magnetic order can be suppressed by tuning parameters? Kitaev QSL survives when suppressing the magnetic order?
23
Longitudinal thermal conductivity κxx
24
Strongly anisotropic response: Quasi 2D nature of magnetic properties.
Suppression of the AFM order by in-plane field component.
6
5
4
3
2
1
0
κ xx
(W/K
m)
86420
µ0H|| (T)
θ = 60º 7.0 K 6.0 K 5.0 K 2.5 K
10
8
6
4
2
0
κ xx
(W/K
m)
14121086420
T (K)
µ0H|| = 7 T (θ = 90º)
0 Tµ0H⊥ = 12 T(θ = 0º)
µ0H|| = 7 T (θ = 60º) 12
10
8
6
4
2
0
T (K
)
86420µ0H|| (T)
κxx(H) κxx(T) θ = 45° θ = 60°
θ = 90°
M/H (θ = 90°)
θ
Summary
25
Measurements of thermal Hall effect in a Kitaev magnet candidate α-RuCl3
Perpendicular fields
Striking enhancement of κxy/T with positive sign below T ~ JK/kB
A broad peak at T ~ 0.1JK/kB
Tilted fields
Observation of half-integer thermal Hall conductance for the first time.
Y. Kasahara et al., arXiv:1709.10286 (2017).
Signature of Majorana fermion excitations
Sudden disappearance of the quantum Hall plateau at high field
Evidence for chiral Majorana edge current
Topological phase transition