Strong correlations and electron-phonon coupling in high-temperature superconductors: a quantum Monte Carlo study

(1)

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Strong correlations and electron-phonon coupling in high-temperature superconductors: a quantum Monte

Carlo study

I. Morgenstern, M. Frik, W. von der Linden

To cite this version:

I. Morgenstern, M. Frik, W. von der Linden. Strong correlations and electron-phonon coupling in high-

temperature superconductors: a quantum Monte Carlo study. Journal de Physique I, EDP Sciences,

1992, 2 (4), pp.393-400. �10.1051/jp1:1992152�. �jpa-00246494�

(2)

Classification Physics Abstracts 74.20

Short Communication

Strong correlations and electron-phonon coupling in high- temperature superconductors:

_a

_quantum Monte Carlo study

1.

Morgenstem

(~>~,~), ^M. Frick (~>~), and W. von der Linden (~>~) (~) IILRZ,

clo

KFA J61ich, Postfach 1913, D-5170 J61ich, Germany

(?)

Institute for Theoretical Physics, University of Regensburg, D-8400 Regensburg, Germany (~) ^IBM ^Research ^Lab. Zurich, Saeumerstr. 4, CH-8803 Ruschlikon, Switzerland

(~) ^Institute ^for Theoretical Physics, University ^of

Groningen,

PO-Box 800, NL-9700 AV

Groningen,

The Netherlands

(~) Max-Planck Institute for Plasma Physics, Boltzmannstr.2, D-8046 Garching, Germany

(Received

9 December 1991, accepted in final form 10 Janu

ary1992)

Abstract. We present quantum simulation studies for _a system ^of strongly correlated fer-

mions coupled _to local anharmonic phonons. The Monte Carlo calculations _are based _on _a

generalized version of the Projector Quantum Monte Carlo Method allowing _a simultaneous

treatment of fermions and dynamical phonons. The numerical simulations yield exact results

in the electron-phonon parameter regime relevant for high-( superconductivity, which is not accessible by perturbative methods like Eliashberg-Theory. The class of electron-phonon models covered in the simulations describes superconductivity exhibiting several features of the _new

high-Tc materials.

After the

discovery

of

high

temperature

superconductivity [I]

an enormous effort has been devoted _to the theoretical

understanding

^of ^the _new materials. The

proposed microscopic

models

mainly geared

for the

description

of

strongly

correlated carriers in the

Cu02-planes

are the

single-band

Ilubbard model [2], ^the

Emery

model [3,

4],

the t-J-model [5], ^and ^the Anderson lattice model [6]. ^These ^models

provide

_a reasonable

description

of the

magnetic

and normal state

properties

of the

copper-oxides. They

revealed _many

unexpected

and

exciting properties

of

interacting

fermions in low dimensions. The

question, however,

whether

they

also

describe

high

temperature

superconductivity

_[7] ^{is still}

controversially

debated.

Quantum

Monte Carlo

(QMC)

calculations have been useful in

clarifying

the

physical

fea-

tures of the

proposed many-body

models. Stabilized

algorithms

allow accurate studies of

the

low-temperature properties. Following

these studies there is _no numerical evidence for the

presence of

high-temperature superconductivity

in the

single-band

Hubbard model

[8-12].

This

finding

has been corroborated

by

recent

analytical

work

[13].

For the

Emery

model the situa- tion is _more subtle due _to the

large

parameter space, but

QMC

calculations _so far do not obtain any substantial

signal

for

superconductivity [14-16]

as well.

QMC simulations, however,

show

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394 JOURNAL DE PHYSIQUE I N°4

superconductivity

in the attractive Hubbard model

[Ii].

Thus the numerical evidence grows that the strong electronic

repulsion

present ⁱⁿ the Cu orbitals

might

not suffice to mediate

pairing

and

merely

determines the

unique quasiparticle

features of the

charge

carriers.

The

original pessimistic

attitude towards the standard

electron-phonon coupling

^has been

changing

^and

sufficiently high

transition temperatures do _not _seem

impossible

_[18].

Many

of the

assumptions

_necessary for standard

electron-phonon

calculations _are not

justified

in the

high-Tc materials,

_as there _are the crude treatment or even

neglect

of the Coulomb

repulsion,

the

dirty limit,

the weak

coupling

limit and the

perturbative

treatment of the

electron-phonon coupling

term in

Migdal-Eliashberg theory.

In the

high-Tc

materials also anharmonic effects

are considered

important [18],

^relevant

phononic

and electronic

energies

_are

comparable

and the Fermi-surface is close to

being perfectly

nested. These

points

contradict the basic _assump- tions of

Migdal-Eliashberg theory.

Anharmonic effects have been studied

earlier,

_e-g- for _a

double well

potential [19],

^but on a more

qualitative

level. In the _case of uncorrelated electrons

coupled

to harmonic

phonons

_a

study

of the

validity

of the

Migdal-Eliashberg theory

for strong

electron-phonon coupling

[20] ^has ^led to reasonable results in the normal _state. Here,

however,

we are interested in

strongly interacting

fermions

coupled

to anharmonic

phonons

in the

superconducting regime.

To

study

_a

generic electron-phonon

model

containing

the essential features of

high-Tc

_super-

conductivity,

_we

performed large

scale

QMC

simulations. Guided

by

the

similarity

bet>veen

high-n superconductors

and

high

temperature ferroelectrics Miiller

suggested

an essential role of the anharmonic _apex _oxygen mode

[21].

^There ^is

growing

evidence

[11,

22, 23] ^for ^the va-

lidity

of this

hypothesis:

_e-g- the universal correlation between Tc and the

Madelung potential

difference between

in-plane

and _apex _oxygen sites

[24],

^the

reported

decrease of the bond

length

of the _apex _oxygen below Tc [25] ^and ^the

pressure-dependence

of Tc in the

T*-phase [26].

Other external

degrees

^of ^freedom ^have ^been

proposed

to mediate the

pairing:

local

charge-

transfer excitations

[27],

copper d-d excitations [28], ^etc..

Although

_we consider the _apex oxygen motion _as

crucial,

_we

emphasize

that the

particular

model Hamiltonians _are _very similar and differ

merely

in the

physical origin

and

magnitude

of the parameters. The modes

coupled

to the carriers have local character and _are

representable

in _terms of

Two-Level-Systems (TLS) [29].

TLS _can _even be

regarded

as an

approximation

to harmonic

phonons.

Our numerical

results

apply

to all these

interpretations.

Turning

towards the electronic structure of the

high-Tc

oxides close to the chemical

potential

in the

undoped

_case, the _oxygen

2p

orbitals _are filled and _on each _copper site _one hole resides in _a 3d orbital

forming

a local

spin _[30].

The additional holes introduced

by doping

have

pronounced

_oxygen character _as _seen in resonant

photoemission

_[31] and

QMC

simulations of the _pure

Emery

model

[14].

^In ^the

following

we assume that the crucial effect of the strong

Coulomb

repulsion

_on the _copper orbitals is the

dynamical separation

of the local

spins

_on the _copper sites and the itinerant fermions in renormalized bands of

predominantly

_oxygen

character. This leads to the model Hamiltonian

7i " -I

~ _(cj~c;,~

₊

_h.c.]

₊ _U

~j

n; in;

(ii') _a j

+g

~ ~

n;+A,a) ^S)

^Q

~j(Sill

#

S[

+ COS #

S(). I)

I A,a I

In this notation

j

labels oxygen sites and I TLS sites above the _copper atom, ^while ^1h denotes the four oxygen sites of the

Cu04-plaquette. (jji)

restricts the

hopping

to nearest

neighbor

sites.

c)~,c;a

^create ^and ^annihilate ^the ^carriers ^with ^spin « at site j. _n;a ₌

cj~cj~

^is ^the

occupation

number. The TLS _are

represented by

Pauli

spin

operators

s$

_v ₌ _{~, y,} _z

). They

(4)

are

coupled

with

strength

_g to the

in-plane

carrier

density

_on the

corresponding plaquette.

The Hamiltonian includes _a kinetic

ax

₌ Q _cos

#

and _a

potential

TLS term Qz ₌ Q sin

#).

It describes the most

general coupling

of the TLS to the carriers in the

plaquette

_geometry

and therefore

generalizes previous

models

[29].

^We stress that the model Hamiltonian does _not contain _a

coupling

of the O carriers _to the Cu

spins.

In terms of the

spin-fermion

model [6], the

exchange coupling

between the itinerant O carriers and the localized Cu

spins

has been

neglected.

The

hopping

matrix element _t _can be estimated _as 0.1- 0.2 eV from cluster calculations [4]

and

angular

resolved

photoemission [31, 32].

For the Hubbard parameter we take _as _a

typical

value U

= 6 in units oft- The

frequency

of the apex oxygen mode has been determined

by

Raman measurements [33] as 50 -100 mev. We chose _g of the _same order of

magnitude.

Performing QMC

calculation for _a wider _range of parameters _we ^found that the

qualitative physics,

in

particular

the _presence of

superconductivity,

is rather insensitive to the choice of the TLS parameters [34].

Furthermore the short coherence

length

indicates that the

energies

of the two

sub-systems

fermions and

phonons

_are of

comparable

size. This leads to the

important

consequence that the Hamiltonian _can not

necessarily

be cast into _an effective attractive electronic interaction like in standard BCS

theory. Secondly,

the

paired particles

_are in rather close distance and

therefore the Coulomb interaction has to be accounted for

properly. Presently QMC

simulations

are the

only

_means to obtain reliable results.

We

employed

the

Projector Quantum

Monte Carlo

(PQMC)

scheme

[35-37]

to obtain _prop- erties of the

ground

state of Hamiltonian

(I).

The fermionic

PQMC algorithm

^has been

generalized

to the electron-TLS _case

using

_a world-line-like

technique

for the TLS

[iii.

We concentrate _our studies _on closed shell _cases, which _are favorable _as far _as the convergence

properties

of the simulation and finite size

scaling

_are concerned

[38].

Details of the numerical

properties

of the present simulations have been

published

elsewhere

[39].

The presence of

superconductivity

is studied in _terms of the

two-particle density

matrix _or rather the

Cooper pair

correlation function

(CPCF)

_[40]. A

macroscopic

quantum state

(su- perconductivity)

is indicated

by

the _appearance

of.Off-Diagonal Long Range

^Order

(ODLRO)

in the CPCF

Xm(1) ) _~

((Cj+m /2i~~-m/21~j+1-m/2i ~j+I+m/2i)

j

~~~~+m/2i~j+1+m/2i~(~j-m/2i~j+1-m/21)) ^(~)

Here _m denotes the distance between the carriers within the

Cooper pair

and I the distance between the

Cooper pairs. Only singlet pairing

is considered.

(..

denotes the

ground

state

expectation

value. ODLRO is present if

xm(I) approaches

_a finite

limiting

value for 1- _c1o.

Quasiparticle

renormalization effects _are eliminated

by subtracting

the

one-particle

contribu- tions in

(2) [41].

^The

integrated

CPCF _are defined _as _xm

=

£j xm(I).

Figure

I shows the

decay

of the CPCF with the

Cooper pair

distance for _s-wave

pairing

of

particles

_on

neighboring

sites, _m ₌ I. Results _are shown for the

electron-phonon

model

(upper part)

and the _pure Hubbard model

(lower part)

for _a 16 x16 lattice with _a carrier concentration of about 15 il.

In the

electron-phonon model,

the CPCF

clearly

reaches _a

positive plateau

for

larger Cooper pair

distances. The results for the _pure Hubbard model _are

substantially

different. The CPCF is

negative

for all distances _a

sign

for the

repulsion

^between the

Cooper pairs

and it

decays exponentially

to zero with distance

(lower inset).

There is also _no

significant signal

for

(5)

olo

l 5

1-Q " 1

~

l~

~

-i~-,---~

/~

,

² ³ ⁵ ^~ ^? ^~

(

_a _~m * "mmm~*

O

~m

~

.

;...'

-o ~

fl-10

-1 . -12

,

. T

ii

,----,---~

l 2 3 5 6 7 8

-1

~ 4 5 6 7 8

Fig.

I. Decay of the nearest-neighbor CPCF _xi (I)

s-wave)

with the Cooper pair distance for the

electron-phonon model

(upper part)

and the Hubbard model

(lower part).

Insets: semi-logarithmic plots. Solid lines _are least _square fits. The

error bars _are of the size of the symbols. Parameters:

16 x 16 lattice, _g

= I-o, Qx= o.5, Qz= o.5, ^U ₌ 6.o, 18 holes.

superconductivity

for other

symmetries.

In _our _case the

coupling

to the TLS is _necessary for

superconductivity.

The _upper inset of

figure

I shows that the CPCF of the

electron-phonon

model decreases

exponentially

for short distances before

leveling

off at _a _cross-over distance of _a few lattice

spacings,

which turns out to be

independent

of the lattice size [29]. ^This ^short cross-over

length

in

particular

^makes

high

temperature

superconductivity

accessible to

QMC

simulations.

A remark is

noteworthy

in this context. The

IIohenberg

theorem [42] ^does ^not

apply

to _a

single

quantum state but

only

to the

thermodynamic

ensemble. In

particular

it does not rule out the existence of ODLRO in _any of the

low-lying

^excited states. This leads _to

a crucial difference between the

meaning

of the

projection

parameter e in the

PQMC

scheme [39] ^and the inverse temperature

fl

in

partition

function MC schemes. Whereas in the

thermodynamic

ensemble

superconductivity

_can

only

be

expected

for

fl

- c1o with

increasing

system

size,

this is not

necessarily

the _case for the

PQMC method,

where

no thermal fluctuations have to be

overcome.

Figure

2 reveals

important

technical details of the simulation for the

electron-phonon

model.

The _average

sign

_[39] is still

substantially high

for

larger projection

parameters b where the saturation

regime

of the

expectation

values here the CPCF is

already

reached.

Although

we find _an

exponential decay

of the _average

sign

with b

(see inset) [43],

we are able to

perform

stable

QMC-simulations

for the parameters of interest.

We

study

_a low band

filling,

which is

given by

the amount of carrier

doping

in the _supercon-

ducting regime.

Here the effects of the Coulomb

repulsion

_on the normal state

properties

_are rather

marginal. However,

the simulations showed that the formation of _a

superconducting

state in the above model is

heavily

influenced

by

the Coulomb

repulsion.

There is

a critical

coupling strength

^below ^which ^the ^Coulomb

repulsion

_supresses

superconductivity

_[34]. This behavior is traced back to the local nature of the

coupling

of the electrons to the

TLS,

_as well

as the

comparable

_energy scale of electrons and TLS.

(6)

,o

Q-B

0.6

<"""""' fl

0.4

u~

0l

no 0

I _~

°° " °°'~~'

0l Z -O 2

-0.4

)-°4

^°°°~

g _~ 6 /

I

-O 8

(~

~~~~

~

~ klfi'l'il'l

20 ~ ~~i~fi'lf120

e a

~~'

0 0.4C6 o-BIG 1.2i[6

blc

sLci~s

Fig.2.

Average sign _as function of Monte Carlo steps for projection _parameters e

= 8

(dashed)

and 16

(solid).

Insets show the average sign _versus e

(a)

and the convergence of the integrated CPCF xi with the projection parameter e

(b).

Parameters _see figure I.

We note that in _a

describing

_an on-site

coupling

of TLS to the carriers [9, II,

44] unrealistically large coupling strengths

and TLS

frequencies

_were _necessary to obtain

a

significant

effect as

superconductivity

is concerned. The introduction of the _more realistic

coupling

to the

Cu04 Plaquettes

_as described in Hamiltonian

(I)

allows _an

experimentally acceptable

set of parameters.

Figure

3 shows the CPCF

(extended

_s-wave

symmetry)

_versus distance _m between the

paired particles picturing

the

spatial

structure of the

Cooper pairs.

The coherence

length

is estimated

a few lattice constants in agreement with

experimental findings.

This result is _a consequence of the correlation effects

leading

to _a _narrow electron band and the

coupling

to the

dispersionless high-energy

TLS mode.

Figure

4 shows the

dependence

of the

integrated

CPCF _xm _on the

frequency

^Q

allowing

a

qualitative study

of the

isotope

^effect

[22].

^The _square ^root ^{of the}

integrated

CPCF

yields

a measure for Tc _as it coincides with the order parameter in BCS

theory. Strictly speaking

the critical temperature in 2D should be _zero due to thermal fluctuations

[42],

but the two dimensional CPCF _governs the transition temperature in systems with weak

coupling

into the third dimension. The

isotope

_mass M enters the model via the TLS

frequency

Q. The

dependence Q(M)

_can be determined from Raman

scattering experiments [33].

For small Q

a considerable

slope

in the

x(Q)-curve

indicates _a substantial

isotope effect,

whereas in the intermediate

regime

the

isotope

effect

approaches

zero in the

vicinity

of the maximum and _can

even

change sign.

In _more traditional

electron-phonon

systems ^the

phonon frequency

is _very small in

comparison

to the electronic band width. This

yields

_a

large isotope

effect

according

to

figure

4.

In summary we

presented

_a model for

high-temperature superconductivity describing

the

coupling

of the

strongly

correlated

charge

carriers within the

Cu02-planes

to local

phonons.

In contrast to

QMC

calculations for

purely

electronic

models,

_we find clear evidence for the presence of

superconductivity.

The model reflects the

experimentally

short coherence

length.

The

isotope

effect in

high-Tc

materials _can be understood in this framework.

(7)

lo

o

~

fl o ~~

*lo~'

~----m__

/

j~°

~lo~' / ~

/

~l

iffy

& B = ~

IIl

x

Fig.3.

Spatial structure of cooper pair. Integrated CPCF _xm _as _a function of the pair extension

m =

(mx, my).

Parameters _see

figure

I.

~iu ^~

~

^o

.0

~~

0. 0.4 1.2 1.6 2.0

ii

FigA.

Integrated _s-wave CPCF _xi _squares and _x2 circles _as function of the TLS frequency

Q. Solid lines _are guides to the _eye. Parameters: 8 _x 8 lattice, _g ₌ 1.0, ax ₌ Qz, U

= 6.0, lo holes.

Acknowledgements.

We _are indebted to K.A. Miller for

bringing

_our attention to the

importance

of anharmonic modes in the

high-Tc

materials and for _numerous discussions. We thank H. de Raedt in

partic-

ular for his support in

developing

the

algorithm. Furthermore,

_we mention useful discussions with A.

Baratoff,

J. G.

Bednorz,

D. Bormann, ^H.

Eschrig,

P.

Fulde,

H.

Herrmann,

II.

Homer,

P.

Borsch,

D. M.

Newns,

P. C.

Pattnaik,

C.

Rossel,

T. Schneider and D. Stauffer. P. C. Pat- tnaik and D. Shea _are

acknowledged

for their

help

_on the 256 V ^' Victor'

Transputersystem

(8)

at the IBM T. J. Watson Rearch Center where

preliminary

calculations _were carried out. I.

M. would like _to thank II. G.

Matuttis,

F. Wfinsch and J. M.

Singer

for their

help

at the

University

of

Regensburg.

He also

aknowledges

the support of the

Aspen

Center for

Physics.

Part of the work

(M. F.)

^has been founded

by

the research _program of the

'Stiching

FOM' which is

financially supported by

the 'Nederlandse

organisatie

voor

wetenschappelijk

anderzoek

(NWOI'.

The present collaboration is part ^{of the}

large

scale

project

^' Numerical Simulations of

High-Tc Superconductivity'

at the

Supercomputer

Center HLRZ

Jfilich, Germany.

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