1 National Institute of Advanced Industrial Scienceand Technology (AIST)2 Hakodate National College of Technology

Theoretical Study ofHigh Temperature Superconductivity

T. Yanagisawa1, M. Miyazaki2, K. Yamaji1

1. Introduction2. Superconductivity3. High Temperature Superconductivity4. Hubbard Model5. Variational Monte Carlo method6. Stripes in high-Tc cuprates7. Spin-orbit coupling and Lattice distortion8. Summary

Outline

1. IntroductionKey words: Physics from U (Coulomb interactions)

•A possibility of superconductivity Superconductivity from U

•Competition of AF and SC

•Incommensurate state Stripes and SC Compete and Collaborate

•Stripes in the lightly-doped region

•Singular Spectral function

0

100

200

300

T(K)

Concentration x in Ln2-x

MxCuO

4-y

SC

AFAF

SC

Hole dopedElectron doped

0.3 0.2 0.1 0 0.1 0.2 0.3 0.4

Pseudo-gap

Stripes

1. Origin of the superconductivity• Symmetry of Cooper pairs• Mechanism of attractive interaction

Coulomb interaction U, Exchange interaction J

2. Physics of Anomalous Metallic behavior• Inhomogeneous electronic states: stripe• Pseudogap phenomena• Structural transition LTO, LTT

Purpose of Theoretical study

2. Superconductivity

Temperature K

1911Kamerlingh Onnes Elements that become superconducting

Superconductive at low temperatures

Superconductive under pressure

Resis

tivity

Characteristics of Superconductivity

Electrical Resistivity

Meissner effect

Metal Semiconductor Superconductor

Exclusion of a magneticfield from a superconductor

3. High Temperature Superconductors

0

20

40

60

80

100

120

140

160

1920 1940 1960 1980 2000

Tran

sitio

n te

mpe

ratu

re (

K)

Year

PbHg

NbNbN

V3Si

Nb3Sn

Nb-Al-Ge

Nb3Ge

K3C

60

MgB2

La2-x

BaxCuO

4

La2-x

SrxCuO

4

YBa2Cu

3O

7

Bi2Sr

2Ca

2Cu

3O

y

Tl2Ba

2Ca

2Cu

3O

y

HgBa2Ca

2Cu

3O

y

HgBa2Ca

2Cu

3O

y(Under Pressure)

TlBaCaCuO

Liquid N2

Liquid H2

Critical Temperature

La2CuO4 YBa2Cu3O7

Phase Diagrams

Phase diagram of HTSC

Electron densityHole density

超伝導

反強磁性

Tc

TN

AF

Superconducting

Carrier density

T

Anomalous metal

Two-Dimensional Plane

Characteristics

•　Two dimensional•　Low spin 1/2•　O level is very closed to Cu level.

Cu

O

dx2-y2

εd

εd+Udεp

Doped hole

px

Model of HTSC

Temperature dependence of Resistivity

H. Takagi et al. T. Ito et al.

Specific Heat

Loram et al., Phys. Rev. Lett. 71, 1740 (1993)

Loram et al., Physica C162-164, 498 (1989)

LSCO

Nuclear Magnetic Resonance

Yasuoka et al., Physica B199 (1994)278

YBa2Cu4O8 　Tc= 81K

Tc

χ

The decrease of 1/TT1 above Tcsuggests the existence of the pseuo-gap.

In the conventional case

4. Hubbard Model

Itinerant Electrons

↑ ↑↓ ↓Atoms

Electrons

Mott insulatorsMnO, FeO, CoO, Mn3O4, Fe3O4,NiO, CuO

Insulators due to the Coulomb interaction

(Note: Antiferromagnets such as MnO and NiO are not Mott insulators

in the strict sense.)

On-site Coulomb Interaction

↑

Coulomb interaction

↑

ε0

ε0+U

U >> t 　Insulator

↑

↑t

+U

↑ ↑

Gap in the Hubbard ModelHartree-Fock theory (Half-filling)

AF Gap Δ = Um Δ ~ t e-2πt/U d = 1, 3 ~ t e-2π(t/U)1/2 d = 2

1D Hubbard modelU << t U >>t

Hubbard gap Δ U(16 / π ) tUe− π / (2U )

Spin-wave velocity 2vs/π = J

�

(4 t /π )(1−U /4πt) 4t2/U

Δε

k0

vsk~U

~1/U

vs

Δ

Cu-O2 Model and Hubbard Model

Cu-O2 model(d-p model)

t-J model Hubbard model

tpd << Ud-(εp−εd)tpd << εp−εdεp−εd << Ud

U >> t

εp−εd ~ 0(tpd)

Zhang-Rice singlet Mixed state of d and p

H = −t (ciσ+

ij σ∑ c jσ + h.c.) + J Si

ij∑ ⋅ Sj H = −t (ciσ

+

ij σ∑ c jσ + h.c.) +U ni↑

i∑ ni↓

5. Variational Monte Carlo methodWe evaluate the expectation values using the Monte Carlo method.

Gutzwiller function

�

ψG = PGψ 0

�

PG = 1− (1− g)n j↑n j↓( )j∏

�

ψ 0 : trial wave function　Fermi sea, AF state, or BCS state

Gutzwiller operator

�

0 ≤ g ≤ 1

Control the on-site correlation in terms of g

weight g weight 1Coulomb +U

Variational Monte Carlo Method

Normal state

�

ψ 0 Slater determinant

Wave numbers: k1, k2, …, kn　Coordinatepositions:j1, j2, …, jn

�

det D↑ =eik1 j1eik1 j2 L eik1 jn

Leikn j1eikh j2 L eik n jn

�

ψ 0 = all∑ ψ l

�

ψ l :particles in the real space

Weight of this state

�

al = det D↑ det D↓

Slater determinant

The large number of particle configurations　→　Monte Carlo method

Monte Carlo algorithm

Expectation value

�

ψQψ = ammn∑ an ψ mQψ n =

am2

al2

l∑m

∑ anam

ψ mQψ nn∑

�

ψ m

�

Pm =am2

al2

l∑

�

ψQψ =1M

anam

ψ mQψ nn∑⎛ ⎝ ⎜

⎞ ⎠ ⎟

m∑

�

m = 1,L ,M

Metropolis法 ψ j → ψ n

R = an2 / aj

2≥ ξ , adopt

�

ψn

< ξ ψ jξ: random numbers 0 ≤ ξ <1

The appearance rate of is proportional to in M.C. steps,

Ifagain

ψ CdS = PG (ukk∏ + vkck↑

+ c− k↓+ ) 0

Gutzwiller Projection PG

To control the on-site strongcorrelation

Weight g Weight 1Coulomb +U Parameter 0<g<1

Equivalent to

RVB state (Anderson)

Superconducting state

k -k

SC Condensation energy

ΔESC = Ωn − Ωs = (Sn0

T c

∫ − Ss )dT

= (Cs0

T c

∫ − Cn )dT

Loram et al. PRL 71, 1740 (‘93)

T

C/T

SC Condensation energy ~ 0.2 meV

optimally doped YBCO

Entropybalance

Superconducting condensation energy

Evaluations in the superconducting state

-0.738

-0.736

-0.734

-0.732

-0.730

0 0.05 0.1 0.15

E g /N s

Δ

0

0.001

0.002

0.003

0.004

0 0.005 0.01

ΔE/N

1/Na

Variational Monte Carlo method10x10 Hubbard model U= 8

YBCO

Econd ~ 0.2meV

SC condensation E

T. Nakanishi et al. JPSJ 66, 294 (1997)

K. Yamaji et al., Physica C304, 225 (1998) T. Yanagisawa et al., Phys. Rev. B67, 132408 (2003)

Condensation energyEcond ~ 0.00038tdp = 0.56 meV/site

δ = 0.333tpp= 0.2

SC order parameter

2D d-p model 6x6 and 8x8

Hole density

Condensation Energy for d-p model

T.Yanagisawa et al., PRB64, 184509 (‘01)

Superconductivity and Antiferromagnetism

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2

SDW(U=6,L=10)SDW(U=6,L=12)SC(U=6,L=10)SC(U=6,L=12)

E cond

t'

Fig. 1

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0 0.002 0.004 0.006 0.008 0.01

Fig. 2SC Econd (ρ ~.84, U=5, t'=−.05)

E cond

1/Ns

Size dependence ofSize dependence ofSC condensation energySC condensation energy

CompetitionCompetition

Pure d-wave SC

0.41meV

Experiments　0.26 meV/site 0.17~0.26 （critical field Hc） (C/T)

AFSC

t’

6. Stripes in high-Tc cuprates

M.Fujita et al. Phys. Rev.B65,064505(‘02) S.Wakimoto et al. PRB61, 3699(‘00)

• Vertical stripes for x > 0.05• Diagonal stripes for x < 0.05

SC+Stripes Coexist

Vertical

Diagonal

Neutron scatteringAF coexists with SC?

T.Y. et al., J.Phys.C14,21(‘02)

Vertical Stripes in the under-doped region

-10

-8.0

-6.0

-4.0

-2.0

0.0

0.2 0.25 0.3 0.35 0.4 0.45 0.5E/N

tpp

Stripes

Commensurate AF

Vertical stripes: 8 lattice periodicity (Tranquada)

x = 1/8Charge rich

Chargepoor

3-band HubbardVMCCharge

rich

8 lattice period

Stripes and Superconductivity

Hij↑ + Fij

Fji* − H

ji↓

⎛

⎝ ⎜

⎞

⎠ ⎟ ujλ

vjλ

⎛

⎝ ⎜

⎞

⎠ ⎟ = Eλ ui

λ

viλ

⎛

⎝ ⎜

⎞

⎠ ⎟

αλ = uiλ a

i↑+ vi

λai↓+

αλ = uiλ a

i↑+ vi

λai↓+

ψ SC = PGPN eαλ

λ∏ αλ

+ | 0 ∝ PG (U−1V)ij ai↑+ aj↓

+

ij∑

⎛

⎝ ⎜ ⎞

⎠ ⎟ N e / 2

|0

Bogoliubov-de Gennes eq.

Wave functionVλj = vj

λ U( )λj = u jλ

↑

↑↓

SC coexists with stripes (AF) Compete and Collaborate

Nano-scale SC

LaLa2-x2-xSrSrxxNiONiO44LaLa2-x2-xSrSrxxCuOCuO44LaLa2-x-y2-x-yNdNdyySrSrxxCuOCuO44

Diagonal stripes are observed for

Diagonal stripes in lightly doped region

0

100

200

300

T(K)

Concentration x in Ln2-x

MxCuO

4-y

SC

AFAF

SC

Hole dopedElectron doped

0.3 0.2 0.1 0 0.1 0.2 0.3 0.4

δ can be explained by 2DHubbard model.

δ = x

Verticalstripes

Diagonal stripes

Incommensurability: Comparison with Experiments

U=8.0 t’= -0.2

LTT,LTO,LTLO,HTT

M.Fujita et al. Phys. Rev.B65,064505(‘02)

Structural transitions: Lattice distortions

Stripes: suggested by Incommensurability

N.Ichikawa et al. PRL85, 1738(‘00)

LTO

LTT

HTT

Hole density

T

LTLO

δ~x

Stripes and Structural transition

1. Anisotropy of the transfer integrals Anisotropic electronic state vertical stripes Diagonal stripes x<0.05 2. Spin-Orbit Coupling induced from lattice distortions 3. Electron-phonon interaction

LTO

LTT

HTT

Hole density

T

LTLO

vertical

What happens under lattice distortions?

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Δ(En-Ecoexist

) / Ns

Δ, (En-Ecoexist) / Ns

β

LTT structural transitions stabilize stripes.• = ty / tx

• X

• Y

One-band Hubbard model (Miyazaki)

Anisotropy of the transfer integrals in LTT phase

tx

LTLO LTTLTTLTLO

LTO

LTT

HTT

Hole density

T

M. K. Crawford et. al.

tx

ty

LTLO

Mixed phase ofMixed phase ofLTT and LTLOLTT and LTLO

Stripes // tilt axisStripes // tilt axis　　

Hole richdomain

Hole poordomain

Stabilize stripes

Possible Stripe Structure 1

tx

LTT HTTHTTLTT

LTO

LTT

HTT

Hole density

T

M. K. Crawford et. al.

tx

ty

LTLO

Mixed phase of LTT and HTTMixed phase of LTT and HTTStripes perpendicular to tilt axisStripes perpendicular to tilt axis

Stable H. OyanagiA. Bianconi

Charge poorCharge rich

-0.1

0

0.1

0.2

0 0.02 0.04 0.06 0.08 0.1

LTT-HTTParallel //stripesPerpendicular

Δ E/N

u

Oscillation of tilt angles

Possible Stripe Structure 2

HSO = ξ(r)L ⋅ S

dxz(r)↑|HSO|d yz(r)↑ = −i2ξ

dyz(r)↑|HSO|d xz(r)↑ =i2ξ

px(x − a / 2, y)↑|Hdp|d xz(r)↑ = −t xze−ik x / 2⋅a

py(x, y− a / 2)↑|Hdp|dyz (r)↑ = −tyze−iky / 2⋅a

Effective iξ term for p-p transfer

Friedel et al., J.Phys.Chem.Solids 25, 781 (1964)Spin-Orbit Coupling induced by the Lattice distortion

Five orbitals × (↑↓):（dx2-y2, dxz, dyz, px, py）

dx 2 − y 2 (r)↑|HSO|d yz(r)↓ =i2ξ

dx 2 − y 2 (r)↑|HSO|d xz(r)↓ =12ξ txz, tyz ≠0 ~ tilt angle

Ｑ1

CuOxygen

Ｑ２

Tilting

7. Spin-orbit coupling and Lattice distortion

Hkin = − (t ij +ijσ∑ icσθij )diσ

+ djσ(Bonesteal et al.,PRL68,2684(‘92))

One-band effective model

(π/2,π/2)

d-p model

Zone boundary

K.Yamaji, JPSJ 57 (1988) 2745. T.Y. et al., JPSJ 74 (2005) 835.

ξ = 0.4

-8

-6

-4

-2

0

2

4

6

8φ = π/100φ = 3π/100Ek=-2(coskx+cosky)

Ek

k(0,0) (π,0) (π,π) (0,0)

Dispersion in the presence of spin-orbit coupling

φ

φ

φ

−φ

−φ

φ

φ

φ

Excitation: Dirac fermionLinear dispersion

Fermi point (half-filling)

Small Fermi surface

Ek

E(kx ,ky) = ±eiφ / 4eikx + e− iφ / 4eik y

+eiφ / 4e−ik x + e−iφ / 4e−ik y

φ

Inhomogeneous d-density wave

Flux state

Flux statePseudo-gap

Gσ (r, r' ,iω) =ϕσm(r)ϕσm

* (r' )iω − Eσmm

∑

Hϕσm(r) = Eσmϕσm(r)

Nσ (k,ε ) = −1πImGσ (k,ε + iδ )

Density of states

Eigenfunction ϕσm(r)

An origin of pseudo-gap

LTO

LTT

HTT

Hole density

T

LTLO

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

φ = 0φ = π/100φ = 3π/100

N(ω

)

ω/ t

Pseudo-gap in the density of states

φ

φ

φ

−φ

−φ

φ

φ

φ

φ

Φ = 4φDiagonal Stripe & d-density wave

Diagonal stripes with Spin-orbit coupling

-0.575

-0.57

-0.565

-0.56

-0.555

-0.55

0 0.02 0.04 0.06 0.08E/N

φ/π

Diagonal

Vertical

Uniformx = 0.03Spin-orbit coupling stabilizes

the diagonal stripes.VMC

bond-centersite-center

Spin-orbit coupling induces flux.

T.Y. et al., JMMM 272 (2004) 183.

d-density waveiΔQY(k) = ck+Qσ

+ ckσ Q = (π ,π )Y (k) = cos(kx) − cos(ky) φ

φ

φ

−φ

−φ

φ

φ

φ

φ

−φ

−φ

−φ

−φ

φ

φ

φ

−φ

−φ

φ

φ

φ

φ

−φ

−φ

−φ

−φ

Inhomogeneous density wave

Stripe

iΔQY(k) = ck+Qσ+ ckσ

Δ lQsσ = ck+lQsσ+ ckσ

k∑

Qs =(π+2πδ,π) verticalQs =(π+2πδ,π+2πδ) diagonal

Nayak, Phys. Rev. B62, 4880 (‘00)Chakravarty et al., PRB63, 094503 (‘01)

d-symmetry

incommensurate

d-density wave

300

200

100

00.300.250.200.150.100.050.00

0

0.001

0.002

0.003

0.004

0 0.005 0.01

ΔE/N

1/Na

ψ CdS = PG (ukk∏ + vkck↑

+ c− k↓+ ) 0

Theoretical estimate ofSC condensation energy

Agreement with Exp.

Econd ~ 0.2meV

IncommensurabilityNeutron scatterings

overdopedunderdoped

YBCO

8. SummaryHTSC and Correlated Electrons

BCS-Gutzwiller functionVariational Monte Carlo study of