Simulating High Tc Cuprates

T. K. Lee

Institute of Physics, Academia Sinica, Taipei, Taiwan

December 19, 2006, HK Forum, UHK

Collaborators

• Y. C. Chen, Tung Hai University, Taichung, Taiwan• R. Eder, Forschungszentrum, Karlsruhe, Germany • C. M. Ho, Tamkang University, Taipei, Taiwan• C. Y. Mou, National Tsinghua University, Taiwan• Naoto Nagaosa, University of Tokyo, Japan• C. T. Shih, Tung Hai University, Taichung, TaiwanStudents:• Chung Ping Chou, National Tsinghua University, Taiwan• Wei Cheng Lee, UT Austin• Hsing Ming Huang, National Tsinghua University, Taiwan

Acknowledgement

Kitaoka and Mukuda, Osaka Univ.

Introduction

To start simulation, need Hamiltonians!

minimal models: 2D Hubbard model –

exact diagonalization (ED) for 18 sites with 2 holes (Becca et al, PRB 2000)

finite temp. quantum Monte Carlo (QMC) , fermion sign problem ( Bulut, Adv. in Phys. 2002)

dynamic mean field theory and density matrix renormalization group, ..

2D t-J type models –

Three species: an up spin, a down spin or an empty site or a “hole”

Model proposed by P.W. Anderson in 1987:t-J model on a two-dimensional square lattice, generalized toinclude long range hopping

jijiji

jijiij nnssJCHcctH

,, 4

1..

iii ccn

Constraint: For hole-doped systemstwo electrons are not allowed on the same lattice site

tij = t for nearest neighbor charge hopping,

t’ for 2nd neighbors, t’’ for 3rd J is for n.n. AF spin-spin interaction

,21spinsi

Introduction• To start simulation, need Hamiltonians! minimal models: 2D Hubbard model – exact diagonalization (ED) for 18 sites with 2 holes (Becc

a et al, PRB 2000) finite temp. quantum Monte Carlo (QMC) , fermion sign p

roblem ( Bulut, Adv. in Phys. 2002). dynamic mean field theory and density matrix renormaliz

ation group, ..

2D t-J type models – no finite temp. QMC – sign problem and strong constrain

t ED for 32 sites with 4 holes (Leung, PRB 2006)

Variational Approach!

Trial wave function:

tr a

,

2

ˆˆ

tr tr

tr tr

a a OO

a

O

a

ap

is configuration indexa is Slater determinant

To calc. a quantity

2

2

a

ap

Variational approach :1. Construct the trial wave function2. Treat the constraint exactly by using Monte Carlo method

Damscelli, Hussain, and Shen, Rev. Mod. Phys. 2003

Phase diagram

holeelectron

Expt. Info.

electron – hole asymmetry!

Phase diagram

Only AFM insulator (AFMI)? How about metal (AFMM), if no disorder?

Coexistence ofAF and SC?

Related to the mechanism of SC?

Important info from experiments

• 5 possible phases:

AFMI, AFMM, d-wave SC, and AFM+d-SC, normal metal.

e-doped system is different from hole-doped..

Broken symmetries:

• electron and hole symmetry

• Afmm, afm+sc and sc all have different broken

symmetries

t’ and t’’ break the symmetry between doping electrons and holes!

Without the long range hopping terms, t’ and t’’, the model Hamiltonian, t-J model, has the particle-hole symmetry: ii cc

From the theory side:

From hole-doped to electron-doped, just change t’/t → – t’/t and t”/t → – t”/t Different Hamiltonians – different phase diagram.

This provides the pairing mechanism! It can be easily shown that near half-filling this term only favors d-wave pairing for 2D Fermi surface!

jiji

ji

jijijiji

ji

jijijiJ

J

CCCCCCCCJ

nnssJH

,,

,

,,,,,,,

,,

,

2

))((2

4

1

In 1987, Anderson pointed out the superexchange term

Spin or charge pairing?

jijiji

jijiij nnssJCHcctH

,, 4

1..

AFM is natural!D-wave SC?

The resonating-valence-bond (RVB) variational wave function pr

oposed by Anderson ( originally for s-wave and no t’, t’’),

BCSCCvuRVB kk

kkk d,,d P0)(P

)1(Pd ii

inn

d-RVB = A projected d-wave BCS state!

The Gutzwiller operator Pd enforces no dou

bly occupied sites for hole-doped systems

AFM was not considered.

2 2

/ , (cos cos )

2(cos cos ) 4 ' cos cos 2 '' (cos 2 cos 2 ) ,

k kk k k v x y

k

k x y v x y v x y v

k k k

Ev u k k

k k t k k t k k

E

four variational parameters, tv’ tv’’ ∆v, and μv

The simplest way to include AFM:

RVB))1(*exp( iz

i

itr Sh

Lee and Feng, PRB 1988, for t-J

Assume AFM order parameters:

zB

zA ssm staggered magnetization

ji ccAnd uniform bond order

jiji

cccc

yjiif

xjiif

ˆ,

ˆ,

Use mean field theory to include AFM,

Two sublattices and two bands – upper and lower spin-density-wave (SDW) bands

Chen, et al., PRB42, 1990; Giamarchi and Lhuillier, PRB43, 1991; Lee and Shih, PRB5

5, 1997; Himeda and Ogata, PRB60, 1999

RVB + AFM for the half-filled ground state (no t, t’ and t’’)

0 02

/

Ne

kkkkkkk bbBaaA

k

kkk

EA

Pd

k

kkk

EB

iiid nnP 1

&

Ne= # of sites

Variational results

)1(3324.0 ji ss

staggered moment m = 0.367

“best” results

-0.3344

0.375 ~ 0.3

Liang, Doucot

And Anderson

bandsSDWupperbSDWlowera kk &

21

22kkkE 2

1222

43 coscos JmkkJ yxk

The wave function for adding holes to the half-filled RVB+AFM ground state

A down spin with momentum –k ( & – k + (π, π ) ) is removed from the half-filled ground state. --- This is different from all previous wave functions studied.

Lee and Shih, PRB55, 5983(1997); Lee et al., PRL 90 (2003); Lee et al. PRL 91 (2003).

0P

)2/1,(

12/

d

01

Ne

kqqqqqqqk

kzh

bbBaaAc

cSk The state with one hole (two parameters: mv and Δv)Creating charge excitations to the Mott Insulator “vacuum”.

Dispersion for a single hole. t’/t= - 0.3, t”/t= 0.2

The same wave function is used for both e-doped and hole-doped cases.

There is no information about t’ and t” in the wave function used.

Energy dispersion after one electron is doped. The minimum is at (π, 0). t’/t= 0.3, t”/t= - 0.2

J/t=0.3

1st e-

1st h+

□ Kim et. al. , PRL80, 4245 (1998); ○ Wells et. al.. PRL74, 964(1995); ∆ LaRosa et. al. PRB56, R525(1997).● SCBA for t-t’-t’’-J model, Tohyama and Maekawa, SC Sci. Tech. 13, R17 (2000)

ARPES for Ca2CuO2Cl2

The lowest energy at

)2/,2/( k

Ronning, Kim and Shen, PRB67 (2003)

Nd2-xCex CuO4 -- with 4% extra electrons

Fermi surface around (π,0)and (0, π)!

Armitage et al., PRL (2002)

)2/1),2/,2/((1 zh Sk Momentum distribution for a single hole calc. by the quasi-particle wave function

< nhk↑> < nh

k↓ >

Exact resultsfor the single-hole ground statefor 32 sites.Chernyshev et al.PRB58, 13594(98’)

a QP state kh=(,0) = ks a SB state ks =(,0), kh=(/2, /2)

Exact 32 site result from P. W. Leung for the lowest energy (π,0) state t-J t-t’-t’’-J

Takami TOHYAMA et al.,

J. Phys. Soc. Jpn. Vol 69, No1, pp. 9-12

(0,0)

QP

(0,0)

SB

(/2,/2)

QP

(,0)

QP

(,0)

SB

a 0.188 -0.0288 -0.0044 0.123 -0.0313

b 0.188 -0.0254 -0.0052 0.159 -0.0085

c 0.202 -0.0302 -0.0005 0.071 -0.002

d -0.273 -0.203 -0.2241 -0.353 -0.1921

e -0.264 -0.195 -0.2154 -0.279 -0.2115

Our Result: (π, 0) is QP at t-J model, but SB for t-t’-t”-J.

(0, 0) is QP for both

QP:Quasi-Particle state

SB: Spin-Bag state

J/t=0.4, t’/t-α/3 and t”/t’=2/3

0P

)0,0(

12/

d

02

Ne

kqqqqqqq

kkZtotalh

h

hh

bbBaaA

ccSk Ground state is kh=(π/2, π/2)for two 0e holes; kh=(π,0) for two 2e holes.

The state with two holes

Similar construction for more holes and more electrons.

The Mott insulator at half-filling is considered as the vacuum state.Thus hole- and electron-doped states are considered as thenegative and positive charge excitations .

Fermi surface becomes pockets in the k-space!

Lee et al., PRL 90 (2003); Lee et al. PRL 91 (2003).

The rotation symmetry of the wave function alternates between s and d symmetry for 2h,4h, 6h,…!

The new wave function has AFM but negligible pairing.An AFM metallic (AFMM) phase but no SC!

d-wave pairing correlation function

2 holes in 144 sites

Increase doping, pockets are connected to form a Fermi surface:

Cooper pairs formed by SDW quasiparticles

three new variational parameters: μv, t’v and t”v

AFMM shows more hole-hole repulsive correlation than AFMM+SC.

The pair-pair correlation of AFMM is much smaller than AFMM+SC.

Difference between AFMM and AFMM+SCThe doping dependence of AFMM is quite

different from AFMM+SC.

AFMM shows more hole-hole repulsive correlation than AFMM+SC.

The pair-pair correlation of AFMM is much lower than AFMM+SC. 12X12

12X12

16X16

16X16

μv, t’v and t”vmv and Δv

mv and Δv

Δv, μv, t’v and t”v

Variational Energy

Phase diagram

k

Qkkyxtriplet cc)kcosk(cos

CT Shih et al., LTP (’05) and PRB (‘04)t’/t=0 t’’/t=0

t’/t=-0.3, t’’/t=0.2

AFMM+SC

AFMM

x0.60.50.40.30.20.1

Possible Phase Diagrams for the t-J model

t’/t=-0.2t’’/t=0.1

H. Mukuda et al., PRL 96, 087001 (2006).

Phase diagram for hole-doped systems

H Mukuda et al., PRL (’06)

The “ideal” Cu-O plane

Extended t-J model, t’/t=-0.2, t’’/t=0.1

0)(P,1 2/,,,d

q

Nqqqke

eCCaCkN

)( 22kkkE Excitation energies are fitted with

Example:

QP excitation

t’/t, t’’/t, and /t are renormalized and “Fermi surface” is determined by Setting =0 in the excitation energy.

spectral weight:

mmk

mmk EECmEECmkA 0

2

,0

2

, 00),(

STM Conductance is related to the spectral weight

Quasi-particle contribution to the conductance ratio

d-RVB (t’=-0.3, t’’=0.2)

d-BCS

E=0.3t for 12X12E=0.2t for 20X20

For a given trial wave function, , we approach the ground statein two steps:

1. Lanczos iteration

||| 1)1( HN

C

|||| 2221)2( HH

N

C

N

C

C1 and C2 are taken as variational parameters

)()( |)(| lnln W H

2. Power Method

)()( |)(|| lim lnln W

nGS H

Beyond VMC approach – the Power-Lanczos method

|

We denote this state as |PL1>

|PL2>

Controversies about pairingOnly t-J (without t’) is sufficient to explain high Tc?

No! --- Shih, Chen, Lin and Lee, PRL 81, 1294 (1997)

64 sites, J=0.4, PL0=VMC, PL1-1st order Lanczos, PL2-2nd order

P. W. Leung, PRB 2006

Summary

• Variational approach has provided a number of semi-quantitative successes in understanding high Tc cuprates : ground state phase diagram and excitaiytons, spectral weight, STM conductance asymmetry, etc..

• Detailed questions about pseudogap, effects of disorder and electron-phonon interactions, etc.. are still to be resolved.

• Finite temp?

Thank you for your attention!b