Hideo AOKI Dept Phys, Univ Tokyo Hideo AOKI Dept Phys, Univ Tokyo Strongly correlated electrons COE...
-
Upload
osborne-mccarthy -
Category
Documents
-
view
229 -
download
0
Transcript of Hideo AOKI Dept Phys, Univ Tokyo Hideo AOKI Dept Phys, Univ Tokyo Strongly correlated electrons COE...
Hideo AOKIDept Phys, Univ Tokyo Hideo AOKIDept Phys, Univ Tokyo
Strongly correlated electronsStrongly correlated electrons
COE symposium “Physics of strongly correlated systems --- from neutron stars to cold atoms”,
Tokyo, 19 Jan 2007
Kazuhiko Kuroki Univ Electro-Commun Ryotaro Arita MPI Stuttgart, now at RIKEN Seiichiro Onari Nagoya Univ
Kazuhiko Kuroki Univ Electro-Commun Ryotaro Arita MPI Stuttgart, now at RIKEN Seiichiro Onari Nagoya Univ
spin
kxky Shiro Sakai Univ. Tokyo
Ryotaro Arita RIKEN Karsten Held MPI Stuttgart
Shiro Sakai Univ. Tokyo Ryotaro Arita RIKEN Karsten Held MPI Stuttgart
Masaki Tezuka Univ. Tokyo Ryotaro Arita RIKEN Masaki Tezuka Univ. Tokyo Ryotaro Arita RIKEN
SC
Ryotaro Arita MPI Stuttgart RIKEN Shiro Sakai Univ. Tokyo
ferromagnetism
Electron mechanism for SC
Orbital degrees of freedom
El-el + el-phonon
HTC
FQHE
Tatsuya Nakajima Tohoku Univ Masaru Onoda Electron Correlation Lab Takahiro Mizusaki Senshu Univ Takaharu Otsuka Univ Tokyo
Tatsuya Nakajima Tohoku Univ Masaru Onoda Electron Correlation Lab Takahiro Mizusaki Senshu Univ Takaharu Otsuka Univ Tokyo
Condensed matterCondensed matter
Electron correlation
SC
F
Fd-electronsd-electrons
s,p-electronss,p-electrons
attraction
phononphonon
el-el repulsion(spin /charge)el-el repulsion(spin /charge)
Tc ~ 0.1ωDTc ~ 0.1ωDTc ~ 0.01tTc ~ 0.01t
anisotropic pairing
10000K 100K100K 10K
isotropic pairing
room T
liq N
Tc (K)
liq 4He
yearorg
anic
s
conventional
oxid
es
© H Aoki
● Weak / strong correlation
● Spin and/or charge fluctuations ( Int’action range short long )
● Internal degrees of freedom ( orbitals )
● Electron and/or phonon mechanisms
● Spin-off to FQHE physics
● Weak / strong correlation
● Spin and/or charge fluctuations ( Int’action range short long )
● Internal degrees of freedom ( orbitals )
● Electron and/or phonon mechanisms
● Spin-off to FQHE physics
Factors governing pairingFactors governing pairing
Hubbard Gutzwiller Kanamori Moriya HTC apres HTC
Studies for correlated electron systemsStudies for correlated electron systems
1960 1970 1980 1990 2000
FLEX (Bickers et al, 1989)SCR (Moriya)RVB (Anderson, …)FRG (Metzner, ..)QMC (Kuroki & Aoki, 1997)VMC (Yokoyama, Yamaji, ..)DCA (Maier et al, 2005)
FLEX (Bickers et al, 1989)SCR (Moriya)RVB (Anderson, …)FRG (Metzner, ..)QMC (Kuroki & Aoki, 1997)VMC (Yokoyama, Yamaji, ..)DCA (Maier et al, 2005)
1. Heavy fermion superconductors spin fluctuation mediated
2. Superfluid 3He hard core interaction
3. SC in the Coulomb gas (GIC, STO?)
Non-phonon mechanism SCNon-phonon mechanism SC
Kohn & Luttinger 1965; Chubukov 1993: Repulsively interacting fermions Attractive pairing channel exists for T 0 (weak, dilute case)
High-Tc cuprate (La2CuO4)
Hubbard model (a generic model)
U tU ~ 5 eV t ~ 0.4 eV
FLEXFLEXDyson’s eq
effective interaction
self energy
self-c
on
sist
en
t lo
op
Attraction SCAttraction SC
V(k,k ’)
Repulsion SC: nothing strangeRepulsion SC: nothing strange
=-VX
X
Repulsion anisotropic SCRepulsion anisotropic SC
+
+
-
-
attraction if has nodes
● Weak / strong correlation
● Spin and/or charge fluctuations ( Int’action range short long )
● Internal degrees of freedom ( orbitals )
● Electron and/or phonon mechanisms
● Weak / strong correlation
● Spin and/or charge fluctuations ( Int’action range short long )
● Internal degrees of freedom ( orbitals )
● Electron and/or phonon mechanisms
Factors governing pairingFactors governing pairing
(Maier et al, Rev Mod Phys 2005)
DCA DCA
hole concentration
d-SC
Uemura plot
T=T F
TF
TC
Tc upperbounded by TC < 0.03t (FLEX, DCA)
Tc ~ TF/100 is VERY low ! Tc ~ TF/100 is VERY low !
for cuprates : TF ~ 10000 K TC < 100K
(Uemura 2004)
Tc
(1)Pairing int’action from el-el repulsion = weak
Cf. Laser-cooled Fermi gasgoes superfulid (Science, 2004)
Tc 0.1 E F
but attractive int’action ↑Feschbach resonance
(3) Pairing from el-el repulsion = anisotropic (i.e., nodes in BCS)
Good reasons why Tc is so lowGood reasons why Tc is so low
(2) Self-energy correction quasi-particles short-
lived
--
+
+
2D or 3D ?2D or 3D ?
(Arita et al, JPSJ; PRB1999; Monthoux-Lonzarich PRB 1999)
kx
ky
kx
kykz
>
Hf
Aoki, J Phys Condens. Matter(2004)
spin fluctuation
ー +
++
Vsinglet :
Vtriplet :
dx2-y2
Qspin+
p, f
+
+
-
++ 2D 3D
singlet triplet
Pairing interactionPairing interaction
Better the nesting, the better for SC ? Better the nesting, the better for SC ?
(Onari et al, PRB 2003)
Im
km
ax
Peak position/width in (k , ) band dispersion Peak position/width in (k , ) band dispersion
(k )(k ) ()()
Why cuprates ?Why cuprates ?
(1) small dp
large teff ~ tdp2/ dp
(2) single-band Hubbard
(Wilson, 1988)
Ag
AuIr
p
d
● Weak / strong correlation
● Spin and/or charge fluctuations ( Int’action range short long )
● Internal degrees of freedom ( orbitals )
● Electron and/or phonon mechanisms
● Weak / strong correlation
● Spin and/or charge fluctuations ( Int’action range short long )
● Internal degrees of freedom ( orbitals )
● Electron and/or phonon mechanisms
Factors governing pairingFactors governing pairing
Vsinglet :
Vtriplet :
charge fluctuationspin fluctuation
ー +
++
↑↓
↑↑
Spin- and charge-fluctuation mediated pairing Spin- and charge-fluctuation mediated pairing
more effective forlonger-ranged repulsion
U=4 t’=0
same peakpositions
differentpositions
n =0.7
n =0.6
n
Phase diagram for the extended HubbardPhase diagram for the extended Hubbard(DCA+QMC: Arita et al, PRL 2004; FLEX: Onari et al, PRB 2004)
V
Utriplet pairing
kx
charge
0
ky
General physical picture:Peak position/height/width pairing symmetry
General physical picture:Peak position/height/width pairing symmetry
0
spin
kxky
(Onari et al, 2004)
Crossover to electron gas Crossover to electron gas
Crossover to electron gas Crossover to electron gas
(B)lattice (half-filled meaningful)(B)lattice (half-filled meaningful)
(Takada, PRB’93)
(A)on-site U extended Hubbard 1/r electron gas (spin fl dominated) (charge fl dominated)(A)on-site U extended Hubbard 1/r electron gas (spin fl dominated) (charge fl dominated)
rs
8.6 3.3s p
dx2-y2
12th neighbour extended H (Onari et al, cond-mat/05)
ps dxy(Takada, 1993)
p s
(A)1/r Hubbard U
dxy dx2-y2p
spin
(Waber & Cromer, 1965)
4d and 5d 4d and 5d
orb
ital ra
diu
s (A
)
atomic #
p
d
Nb4d4 Ag
AuIr
Sr2RuO4Sr
Ru(4d) Cu(3d)
O
q2D Sr2RuO4q2D Sr2RuO4
(Maeno et al )
Pairing symmetry: triplet p+ip(Sigrist & Rice)
px+y
+ i =
px-y
(Arita et al, PRL 2004)Time-reversal broken triplet pTime-reversal broken triplet p
When T-broken pairing can occur? When T-broken pairing can occur?
When the space group of the pair has two-dimensional rep: as in
● p + ip in tetragonal systems
● d + id in hexagonal systems (6+)
(Onari et al, PRB 2002)
d1 d2
+ i
More recent candidate --- Skutterdite RET4X12
Spin: T-reversal Sr2RuO4
Orbit: T-reversal
Spin: T-reversal non-unitary statesOrbit: T-reversal (ie, broken SU(2))
Non-unitary SCNon-unitary SC
● Magnetic-field induced triplet pairing
(Arita, Kuroki & Aoki, JPSJ 2004)
3He ● A1 phase of superfluid 3He
● Ferromagnetic SC(UGe2, etc)
p
T
A1
Solid
Super-fluidAB Liquid
B
FFLO: Cooper pair with a momentumFFLO: Cooper pair with a momentum
a
q
Fulde-Ferrell-Larkin-Ovchinikov
k space real spaceCeCoIn5
Ultrasonic
NMRWatanabe et al
Kakuyanagi et al
T
μB Quark-gluon plasma
~ 1 GeV
~ 170 MeV
Coloursuper-conductor
Hadronic fluid
Vacuum
Outlook (5) Colour superconductivityOutlook (5) Colour superconductivity
●
neutron starFFLO
● Weak / strong correlation
● Spin and/or charge fluctuations ( Int’action range short long )
● Internal degrees of freedom ( orbitals )
● Electron and/or phonon mechanisms
● Weak / strong correlation
● Spin and/or charge fluctuations ( Int’action range short long )
● Internal degrees of freedom ( orbitals )
● Electron and/or phonon mechanisms
Factors governing pairingFactors governing pairing
Multi-orbital systemsMulti-orbital systems
J: Hund’s coupling
Superconductivity
U - 3J
U Intraorbital
Coulomb
U - 2J
InterorbitalCoulomb
Hund’s coupling
Magnetism
(a) (b)
(c) (d)
(a) Singlet Triplet(b) Triplet Singlet
Cooper pair = (real space)x(spin) x(orbital) = antisymmetric
Hund’s coupling pairing symm DMFT+QMC result (Sakai et al, PRB 2004)
● Weak / strong correlation
● Spin and/or charge fluctuations ( Int’action range short long )
● Internal degrees of freedom ( orbitals )
● Electron and/or phonon mechanisms
● Weak / strong correlation
● Spin and/or charge fluctuations ( Int’action range short long )
● Internal degrees of freedom ( orbitals )
● Electron and/or phonon mechanisms
Factors governing pairingFactors governing pairing
attraction
phononphonon
el-elrepulsionel-elrepulsion
isotropic pairing
Tc ~ 0.1ωDTc ~ 0.1ωDTc ~ 0.01tTc ~ 0.01tanisotropic pairing
Which is better ?or, what if they coexist ?
Anti-adiabatic limit
Adiabatic limit
=
U/t
ω/t
λ/t(Tezuka et al, 2005)
Parameter space inHolstein-Hubbard model Parameter space inHolstein-Hubbard model
U-t
(U,t’)=(2.0,-0.5)
http://buckminster.physics.sunysb.edu/
trestle lattice
cf. A3C60: fccSC dominates!
Break the el-hole symmetry Break the el-hole symmetry
sc can dominate(Tezuka, Arita & Aoki, PRL 2005)
In 1D Hubbard model: degeneracyCDW=SC lifted when el-hole asymm
Ferromagnetism Ferromagnetism
E =
p2 /
2m
s
d
transition metals p
EF
Ferromagnetism very difficult to realise Ferromagnetism very difficult to realise
Repulsively interacting electrons (Hubbard model) Ferromagnetism ?
Stoner (UD(EF) > 1) too crude a criterion
(1) Any rigorous F ? (Nagaoka; Lieb, Mielke, Tasaki)(2) Why are real metallic magnets (Ni, Co, Fe) F ?
Band (itinerant) ferromagnetism Band (itinerant) ferromagnetism
U tU t
15 puzzle15 puzzle
U >> t
He
Multiple-exchange in HeMultiple-exchange in He
Ceperley
Design of a ferromagnetic polymerDesign of a ferromagnetic polymer
polyaminotriazole
N
C・・・ ・・・
Kanamori theory (T-matrix approx.)
χpp = + +… 2,
1 (1 )pppp
UU J
J
UU
negligible
Particle-particle scattering
Particle-hole scattering
2,
1 (1 )pph h
UU J
J
UU
J can be important!
χph = + +…
1-orb. FLEX: Arita et al.’00.
Reliable for low electron densities Ni: 2 holes in 5 orbitals
General band filling
Insulating FM with an antiferro-orbital orderInsulating FM with an antiferro-orbital order
Metallic FM (Ni, Fe, Co) --- how does Hund’s coupling work? Metallic FM (Ni, Fe, Co) --- how does Hund’s coupling work?
?
Itinerant FM in multiorbital systemsItinerant FM in multiorbital systems(Sakai, PhD thesis 2006)
3D fcc lattice with t=4t’=0.28 (Weff=4).Cf.) 1 orb: Ulmke’98
• Lattice structure is important.• U’ suppresses FM.• J enhances FM.• U’ & J are crucial both for n=1.5 and n=0.75 (~Ni).
n=1.5U=4
(i) U’=J=0 (1 orb.)
(ii) U’=4, J=0(iii) U’=3.5, J=0.25(iv) U’=3, J=0.5
U’=U-2J for (ii)-(iv).
Itinerant FM in multiorbital systems - 1st numerical resultItinerant FM in multiorbital systems - 1st numerical result(Sakai, PhD thesis 2006)
Correlated electron systemsCorrelated electron systems
FQHE systemsFQHE systems
ρxy
B
Fractional quantum Hall effect Fractional quantum Hall effect
1/ν (∝ B )
Pan et al 2002
ρxx
0 1 2 3 4
2DEG
ρxyρxxB
Where does the quantum zero point come from ?Where does the quantum zero point come from ?
Liquid He [x, p] = ihLiquid He [x, p] = ih
FQHE [x, y] = ihFQHE [x, y] = ih
H = (1/2m)2, = p +eA
R = (X,Y), [X, Y] = il2
= -(1/eB) ez x , [x, y] = -il2
non-commutative space !
l = (h/eB)1/2: magnetic length ( 80A for B=10T)
FQHE system = Many-fermion system with Coulomb repulsion accompanied by uncertainty in (X, Y)
FQHE system = Many-fermion system with Coulomb repulsion accompanied by uncertainty in (X, Y)
N
S
Composite fermion picture-- flux attachmentComposite fermion picture-- flux attachment
0 0.1 0.2 0.3 0.4 0.5
N =0
N =2
N =1
Compressible liquid
Stripe
Laughlin stateWigner crystal
BubbleDMRG result: Shibata & Yoshioka 2003
1
Triplet p-ip (Pfaffian state 3He A1 Sr2RuO4) Trial wf: Moore-Read, Greiter-Wen-Wilczek 1991
Numerical: Morf 1998, Rezayi-Haldane 2000; Onoda-Mizusaki-Aoki, 2003 Experiment: Willett-West-Pfeiffer 1998, 2002
CF liquid BCS paring at = 5/2 ?
CF liquid BCS paring at = 5/2 ?
CF
CF
B
p
T
A1
Solid
Superfluid
B A
3He
Sr
Ru(4d) Cu(3d)
O Reminds us of Kohn-Luttinger 1965 every metal superconducting with p,d,f,… pairing
T 0
Interaction form Coulomb gauge field range Landau level spin/charge CF interaction
HTC FQHE
Band structure anisotropic isotropic
Summary Summary
© Aoki 2005
composite-boson picture for the Bose-Einstein condensate
(©Kasamatsu et al, 2003)
(Nakajima & Ueda, 2001; 2003;2004)
Magnetar(4 papers in Nature 31 August 2006)
B > 1011 T ?
Summary: correlated electronsSummary: correlated electrons
OutlookOutlook● Relation with neutron star physics Colour SC Ferromagnetism, . . . ● Relation with cold atom physics
● Relation with neutron star physics Colour SC Ferromagnetism, . . . ● Relation with cold atom physics
SC
ferromagnetism
● Lattice models (Hubbard, etc)
● Coulomb gas
● Hard-core system
● FQHE
● Lattice models (Hubbard, etc)
● Coulomb gas
● Hard-core system
● FQHE