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