Extended t-J model – the variational approach

T. K. Lee

Institute of Physics, Academia Sinica, Taipei, Taiwan

July 10, 2007, KITPC, Beijing

Outline• Background and motivation

• Variational Monte Carlo method -- basic approach -- improve trial wave functions increase variational parameters power Lanczos method

• Hole-doped cases, antiferromagnetic (AFM) and superconducting states.

-- ground state -- excitations

• Electron-doped cases

• Summary

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

Phase diagram

holeelectron

Phase diagram

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

Coexistence ofAF and SC?

Related to the mechanism of SC?

H Mukuda et al., PRL (’06)

Phase diagram for multi-layer systemsUD Hg-1245

(Tc=72K, TN~290K)

AFM (0.69μB)

AFM (0.67μB)

AFM (0.67μB)

AFM (0.69μB)

AFM (0.67μB)

AFM (0.67μB)

SC(72K)

+AFM(0.1μB)

SC(72K)

+AFM(0.1μB)

SC(72K)

+AFM(0.1μB)

SC(72K)

+AFM(0.1μB)

“ideal” Phase diagram for hole-doped cupartes?

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

Kitaoka, Mukuda et al. PRL (‘06)

Zero-field NMR spectrum at 1.4K

Coexisting state of SC and AFM is realized on a single CuO2 plane (OP) !!

Nh(IP)~0.06, Nh(OP)=0.22

Nh(IP)~0 , Nh(OP)~0.08?

Nh(IP)~0.08? , Nh(OP)=0.24

AFMM

15.20 15.22 15.24 15.26 15.28

H//coriented powder

Hg- 1245 f=174.2MHzT=280K

Inte

nsi

ty (

a.u

.)

H (T)

OP IP+ IP*

Linewidth 50 Oe (IP)

Flatness of CuO2 plane

150 Oe

50 Oe

Narrowest among high-Tc cuprates !

・ Ideal flatness 　

・ Hole is homogeneously doped

Slide From H. Mukuda

Basic 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: particle and hole; AFM, SC…

Theoretical challenge:the simplest model to account for all these properties?

To start, only consider the homogeneous solution.

Minimal theoretical models

• 2D Hubbard model – U and t differ by almost an order of magnitude, reliable numerical approaches:

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)

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 , t’ and t’’ breaks the particle-hole symmetryJ is for n.n. AF spin-spin interaction

,21spinsi

Minimal theoretical models

• 2D Hubbard model – U and t differ by almost an order of magnitude, reliable numerical approaches

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)

2D t-J type models –

no finite temp. QMC – sign problem and strong constraint

ED for 32 sites with 1,2 and 4 holes (Leung, PRB)

Variational Monte Carlo method

Expectation values of an operator O in |>:

Bases:

-- the probability of config. α

Metropolis algorithmSlater determinant

After a series of configurations (1,2,…,M) is generated, expectation values of the operator O is given by

with error

To estimate the accuracy of <O>, Do MG independent Monte-Carlo runs with different initial configurations.

with error

Here, M total number of configurations! Is it accurate?

Linear optimization method

Variational parameters set {}

Search for the optimized parameters opt which give the minimum energy,

Employ a simple linear optimization of each parameter in 1D with the other fixed. Linear optimization may not work well if correlated trial wave functions require the use of many parameters. Then, what should we do?

Stochastic reconfiguration (SR) method

Optimization for variational energy:

By Taylor expansion of for ,

pi: variational parameters, i=1,2,…,vp.: a given configuration {R1,…,RN}

, where the Oi operator is

Up to O(pi2)

First,

Casula, et. al., J. Chem. Phys. ’04Sorella, PRB ’05Yunoki and Sorella, PRB ‘06

Similar to steepest descent (SD) method, we define a “force”:

The energy improvement is approximately written as

Energy will converge to the minimum when all Fi=0.We can tune t(>0) to control the convergence of iteration.

We can obtain the next parameters pi’ by iterating

Similar to steepest descent (SD) method, we define a “force”:

However, SD method overlooks a possibility that a small pi may lead to a large change of the wave function…

Therefore, we also need to minimize a “distance”:

we ignore O(pi3) terms

A functional is defined as

is a Lagrange multiplier

Minimization of f(pi) with respect to pi, we have a “new” iterated formula:

t=1/2

Then, the energy improvement for SR method becomes

Sij matrix remains positive definite.

Sometimes Sij has no inverse matrix for some unstable iterations. If so, we use the SD method instead.

How do we choose t in SR method?

2D Lattice size=64t-t’-t’’-J model with (t’,t’’,J)=(-0.3,0.2,0.3) Trial wave function: d-wave RVB wave functionn is the number of iterations

linear

SR

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>

J=0.3 E

Exact -0.583813

RVB(PL0) -0.5431

RVB(PL1) -0.5654

RVB(PL2) -0.5709

PL Energy

After PL, nk, HH(R), and energy get closer to exact results!

For the t-J model, LeungPRB 2006, 4 holes in 32 sites

HH(R) – the hole-hole correlation function

Trial wave functions for hole-doped systems

Must account for : Mott insulatorAFM, SC , AFM+SC?

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

Lee and Feng, PRB38, 1988; Chen, et al., PRB42, 1990; Giamarchi and Lhuillier, PRB

43, 1991; Lee and Shih, PRB55, 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 or removing electrons from 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’)

64 sites

Wave Functions for one hole system

• Quasi-particle state:

0)(1

2'

N

kqqqqqqakd bbBaaACP

Hole momentum (kh)=unpaired spin momentum (ks) = k

0)(1

2'

N

kqqqqqqakd

h

sbbBaaACP

)2

,2

(

hs kkwhere ,

• Spin-bag state:

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 new wave function has AFM but negligible pairing.It could be used to represent an AFM metallic (AFMM) phase.

d-wave pairing correlation function

2 holes in 144 sites

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 stronger hole-hole repulsive correlation than AFMM+SC.

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

μv, t’v and t”vmv and Δv

mv and Δv

Δv, μv, t’v and t”v

2k

2kkyxyxyxk

yxkk

kk

k

kk

+=E,)k2cos+k2(cost ′′2kcoskcost′4)kcos+k(cos2=

)kcosk(cos=,E

=u

v=A

Δξμξ

ΔΔΔ

ξ

----

--

All trial wave functions:

RVB(SC): 0)ccA(P̂=SC 2/N

k

+k

+kk

e∑ ↓-↑

)nn1(=P̂ iii

↓↑-∏

AFMM+SC:

0)bbB+aaA(P̂=SC+AFMM 2/N'k

+k

+kk

+k

+kk

e∑ ↓-↑↓-↑

-)2cos2(cos2-coscos4-)cos(cos4

)(,-

,

22

22--

yxyxyxk

kkkk

kkk

k

kkk

kktkktmkk

EE

AE

B

AFMM:0)bbB+aaA(P̂=AFMM 2/N'

q≠k+k

+kk

+k

+kk

e∑ ↓-↑↓-↑

22yxk

k

kkk

k

kkk m+)kcos+k(cos4=,

E=B,

E+=A ξ

Δ

ξ

Δ

ξ -

Variational Energy

Phase diagram

SDW: 0)aaA(P̂=SDW 2/N'k

+k

+kk

e∑ ↓-↑

)(=A kk--ξΘ bandsSDWupperb;SDWlowera kk : : σσ

The energy difference among these states

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

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

Summary – I • Variational approach provides a quantitative way to unde

rstand the t-J or extended t-J model by taking into account the projection rigorously.

• Based on the RVB concept, trial wave functions for the doped system have been constructed to represent the AFMM, AFMM+SC and SC phases observed in multilayer cuprates.

• With the values of t, t’ and J in the range of experiments, the phase diagram obtained agree with cuprates below optimal doping.

Acknowledgement

• 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

Outline• Background and motivation

• Variational Monte Carlo method -- basic approach -- improve trial wave functions increasing variational parameters power Lanczos method

• Hole-doped cases, antiferromagnetic (AFM) and superconducting states.

-- ground state -- excitations

• Electron-doped cases

• Summary

Excitations in the SC state

)cos(cos,/ yxqq

qqqqq qq

Euva

With a fixed-number of holes, the d-RVB state becomes

The ground state 0)(P,,d'

k

kkkkt

CCvuRVB

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

q

Nqqqke

eCCaCkN

)( 22kkkE

Quasi-particle excitation

Excitation energies are fitted with

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.

J/t=0.3, t=0.3eV t’/t=-0.3 & t’’/t=0.2 for YBCO and BSCO

but t’/t=-0.1 & t’’/t=0.05 for LSCO

Our VMC overestimates the gap by a factor of 2, PRB 2006

Anomalies in the spectral weight for the low-lying excitations.

For Gutzwiller-projected wave functions??

For ideal Fermi gas : 0

kk n1Z 0

kk nZ

For BCS theory : 2

kk uZ 2kk vZ

mmk

mmk EECmEECmkA 0

2

,0

2

, 00),(

Exact Identity :

S. Yunoki, cond-mat/0508015, C.P. Nave et al., cond-mat/0510001

No exact relation is known about

Using the identities,

Is this relation between pairing amplitude and spectral weight also true for projected wave functions?

BCS theory predicts

2k

2k

2kkk vuZZ

So what is ?

YES!!

Pairing amplitude by VMC (II):

C.T. Shih et al., PRL

Long-range pair-pair correlation

Anomalies in STS conductance

S.H. Pan et al., unpublished data

OPD

T. Hanaguri et al., Nature

Ca2-xNaxCuO2Cl2

MgB2

McElroy et al. Science (05)

Could the asymmetry be DOS effects ?

d-BCS with 2 bands

B. W. Hoogenboom et al. PRB (‘03)

as hole doped

d-wave gap opened !

DOS singularity

Hoffman

M. Cheng et al. PRB (‘05)

Particle-hole asymmetry for STS conductance

2

, 0 00k m k km k

m C E E Z E E

0k kk

Z E E

the tunneling conductance at negative bias:

For positive bias:

mmk

mmk EECmEECmkA 0

2

,0

2

, 00),(

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

Quasi-particle contribution to the conductance ratio :

d-RVB (t’=-0.3, t’’=0.2)

d-BCS

Spectral weight PEAK HEIGHTS and GAP SIZES

Why is d-wave SC so robust?

McElroy et al. Science (05)

Consider the impurity elastic scattering matrix element

, ' , ', 'k k k kV k C C k

For BCS theory, this is just uk u k’ – vk v k’

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

q

Nqqqke

eCCaCkN

But for projected states, there is a strong renormalization

, ' , ',1, 1, 'k k e k k eV N k C C N k

, ' , ',1, 1, 'k k e k k eV N k C C N k

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

q

Nqqqke

eCCaCkN

gt=2x/(1+x) RenormalizedMean-fieldtheory ,Garg et al,Cond-mat/0609666

15 holes in 12*12

Summary - II

• In addition to studying ground states, variational approach could also study excitation spectra, STM conductance asymmetry, effects of disorder (why d-wave is so robust) etc..

Hole-doped Electron-doped

T’ phaseT phase

Electron-doped systems

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

Phase diagram:

jijiji

jijiij nnssJCHcctH

,, 4

1..

t for n.n., t’ for 2nd n.n., and t” for 3rd n.n.

Consider t-t’-t”-J model:

For electron-doped, no empty sites!! “Lower Hubbard band is projected out”

Experiments M. Ikeda,

thesis ‘06

T. Kubo, Physica C ‘02

VMC

(0.1,-0.05)

(t’,t’’)

(0.15,-0.1)

(0.3,-0.2)

Recent ARPES for Nd2-

xCexCuO4:

16.5 eV --- Armitage et al., PRL 2002

55 eV --- Armitage et al., PRL 2001

(x=0.15)

400 eV --- Claesson et al., PRL 2004

(x=0.15)

22 eV --- H. Matsui et al., 0410388

(x=0.13)

NCCO

SCCO

Non-uniform gap!(x=0.13, TN=110K, T=30K)

x=0.14, Tc=13K,T~18K, TN=110K

• A gap near (π,0) region, but not near (π/2,π/2)• The gap increases as it moves away from (π,0)• The “lower” band approaches Fermi energy near (π /2, π /2) and the peaks are broad• Could these be lower and upper SDW bands??

Near (π,0) Near (π/2,π/2)

t-J model only has the upper Hubbard band

+

“AF LRO”

k and k + Q are coupled!!Q= (π,π)

In weak coupling analogy, 2 “SDW” bands

zB

zA ssm staggered magnetization:

ji ccuniform bond order:

jiji cccc

yjiif

xjiif

ˆ,

ˆ,

d-wave RVB pairing:

2 sublattices and 2 bands – upper and lower SDW bands

G.J. Chen at al. 1990, Giamarchi and Lhuiller, 1991

Same asa

same as the hole-doped case

..

,,

CHbbaa

bbaaH

kkkkkk

kkkk

kkkkMF

bandsSDWupperbSDWlowera kk &

21

222

43 coscos JmkkJ yxk

yxk kkJ coscos43

Mean-field Hamiltonian for lower and upper SDW bands:

with

A down spin with –k (–k+Q) is removed from the half-filled ground state.

Dope one hole

Create charge excitations in the Mott Insulator “vacuum”,The AFMM state:

Dope one electron

0bbBaaA'C P

C)2/1S,k(

12

N

kqqqqqqqkd

0kzh1

0bbBaaA'CCC P

C)2/1S,k(

12

N

kqqqqqqqQkQkkd

0kze1

T. K. Lee, et al., PRB 1997; T. K. Lee, et al., PRL 2003; W. C. Lee, et al., PRL 2003.

all quantities could be calculated with hole wf except t’/t → – t’/t and t”/t → – t”/t

0)(P

)2/1,(

12'

d

01

N

kqqqqqqqkQkQk

kze

bbBaaACCC

cSk

1 electron wave function:

Another similar wave function:

0)(P

)2/1,(

12'

d

01

N

kqqqqqqqkkQk

Qkze

bbBaaACCC

cSQk

These 2 wave functions have a finite overlap!

(0,0) (π,0) (π,π) (0,0)

Fermi level

2 occupied states for the same k

0bbBaaA'

CCCC P)2/1S(

102

N

region ,0)(qqqqqqq

10

region ,0)(kQkQkkkdze20

Consider 20 doped electrons in 144 sites:

or Outside (π,0) region

(0,0) (π ,0) (π,π) (0,0)

Fermi level

Remove an electron from this (π,0) region

(0,0) (π ,0) (π,π) (0,0)

Fermi level

(1)

(2)

0bbBaaA'

CCCC P)2/1S(

102

N

region ,0)(qqqqqqq

10

region ,0)(kQkQkkkdze20

Consider 19 doped electrons in 144 sites:

another similar wave function: )2/1,(19 ze SQk

QP state

0bbBaaA'

CCCCCCC P)2/1S,k(

102

N

k & region ,0)(qqqqqqq

9

region ,0)(kQkQkkkQkQkkdze19

0bbBaaA'

CCCCC P)2/1S,k(

112

N

k & region ,0)(qqqqqqq

10

region ,0)(kQkQkkkkdze19

(1)

(2)

Spectral Weight

(π,0) (5π/6,π/6) (2π/3,π/3) (π/2,π/2)

USDW (19e) 3.23E-01 2.99E-01 2.08E-06 8.61E-08

USDW (27e) 3.08E-01 3.00E-01 9.76E-06 8.31E-08

LSDW (19e) 5.63E-02 7.51E-02 1.41E-02 1.29E-02

LSDW (27e) 1.08E-01 1.12E-01 2.07E-02 1.90E-02

12 by 12, x=013 & 0.19

16 by 16, x=0.17

Gap exists only near (π,0)!Gap is K-dependent!

Conclusions:

• The e-doped t-t’-t’’-J model has been studied in the low energy states.

• At low doping, small FS pockets appear around anti-nodal points.

• Upon increased doping, AF will become weaker.• Based on the AFMM states and coupling between k and k

+ Q= (π,π), the evolution of the FS with doping and the calculated spectral weight are in good qualitative agreement with the ARPES data.

• Questions remained: The doping dependence of phase diagram? Other excited states to be considered? SC+AFM coexistent state ?

Thank you for your attention!