Double layer Hubbard model : off-diagonal long range order in the nodeless d-wave channel

(1)

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Double layer Hubbard model : off-diagonal long range order in the nodeless d-wave channel

I. Morgenstern, Th. Husslein, J. Singer, H.-G. Matuttis

To cite this version:

I. Morgenstern, Th. Husslein, J. Singer, H.-G. Matuttis. Double layer Hubbard model : off-diagonal

long range order in the nodeless d-wave channel. Journal de Physique I, EDP Sciences, 1993, 3 (4),

pp.1043-1051. �10.1051/jp1:1993109�. �jpa-00246768�

(2)

Classification

Physics

Abstracts

74.20

Double layer Hubbard model

_:

off-diagonal long _range order in

the nodeless d-wave channel

1.

Morgenstem,

Th.

Husslein,

J. M,

Singer,

and H,-G. Matuttis

Department

of

Physics,

University ^of

Regensburg,

D-8400

Regensburg, Germany

and HLRZ, c% KFA Jiilich, Postfach 1913, D-5170 Jiilich,

Germany

Abstract. We report on a numerical study of the double

layer

Hubbard model

describing

the

Cu02

sheets in the

high-T~

oxides. For the simulation _we

employed

the

projector

_quantum Monte Carlo method (PQMC) to

study

the

ground

state

properties

of this correlated electron system. Our results

provide

evidence for

off-diagonal long

_range order in the nodeless d-wave channel, The

implications

with respect to

possible

mechanisms for

high-T~ superconductivity

in oxides in combination which phononic booster mechanisms _are discussed.

Introduction.

The

high-T~ superconductors

discovered in 1986

by

Bednorz and MUller

ill

_are still

challenging

condensed matter theorists,

Many physicists

have been

trying

to find _a

pairing

mechanism within the

single

band _one

layer

Hubbard

model,

_yet _« _even _a casual

glance

at the

experimental

facts convinces _one that

superconductivity

is caused

by

^effects outside that

simple

model and does _not _occur with

high

_{T~ in} an isolated cuprate

plane

»

[2]. According

to

P.W. Anderson

«phenomenological theory

and

experiments

indicate that the suitable

electronic _structure model of the

CUO-layers

is that of the _one band Hubbard model _». Its

current

popularity

stems from his

proposal

that electron correlations

might play

the essential

role in

high temperature superconductors [2].

The strong electron-electron correlation

origi-

nates from the

competition

between the kinetic and the Coulomb _energy.

Despite

its

seeming simplicity

and the tremendous effort devoted _to

investigating

the Hubbard

model,

the

physical

consequences, which arise from this

competition,

_are

by

far _not

yet fully

understood.

Numerical simulations based _on the

Quantum

Monte Carlo method

during

^the recent years

mainly provided

evidence

against superconductivity

for this model except ⁱⁿ ^the case of the

« apex oxygen » extension which considers additional

phononic degrees

^of ^freedom

[3, 4].

Until _now there is _no evidence in

single layered purely

electronic models, After first indications shown in _a

previous publication [5]

here _we _want to

give

clear evidence in favour of _an electronic

pairing

^mechanism in two Hubbard

layers coupled by

an on-electron

interlayer

hopping-term

t~

This

interlayer

term

gives

rise to

bonding

and

antibonding

Fermi

surfaces,

_as

already

indicated in the

recently published

_paper ^of Bulut _et al.

[6]. They

discussed

pairing

correlations

(3)

1044 JOURNAL DE PHYSIQUE I N° 4

with _« nodeless d-wave _» symmetry in this type ^of ^model ^and ^concluded ^that ^their ^results

might

be

compatible

^with

experimental findings

which _so far _were

only explained

in terms of

(extended-)

_s-wave

superconductivity,

But

they

_were not able to

give

clear evidence for _a

superconducting

state in their

Quantum

Monte Carlo simulations _as well _as in their

analytic

work.

Model.

The double

layer

Hubbard model is defined

by

the lattice Hamiltonian

~

<l

«

~~"

_~'" _~

~'~'~

_~~

~[~,,

_~~~«

_~>«

+ hC.

)

₊ U

z

n i ⁿ j

,

(i~

mPan~ ~~

where

Qj')

and

iii')

denote nearest

neighbours.

The first

part

^with t describes the

inplane particle motion,

whereas the second term concems a

tunneling

of fermions between

layers

with

a finite

interlayer hopping

term t~. ^The ^matrix

hopping

elements t serves in the

following description

_as _an _energy unit,

Method _:

projector quantum

Monte Carlo.

For the numerical simulation _we

employ

the

projector quantum

^Monte ^Carlo ^method

(PQMC),

which has been proven ^to be well suited for fermionic

groundstate

calculations

[7-11].

The

key

idea of the

PQMC

method is to filter out the

groundstate compound (0)

from _a proper trial function

(1l'~) by applying

_an

appropriate

functional of the Hamiltonian _:

e~ ^~~

V'~)

=

e~ ^~~°

101 V') 10)

₊

Z ^e'~~°~ ^{In1 V'~)} ^In) ⁽²⁾

In this notation

Eo

_<

Ei

_S

E~

denote the

eigenvalues

of 3C and

( in) )

_are excited states.

In the limit fJ- _ct~ this

yields

~j

~ ~- ^ax

~~j j

~

j~j

~~~

For this

algorithm

it is necessary that the

groundstate

is

nondegenerate,

because the convergence of exp

(- p

JC

1l'~)

to the

groundstate

_as a function of the

projection

parameter e will be slow in the _case of

nearly degeneracy.

There is _no way in

performing

the limit fJ

- ct~

numerically,

but

sufficiently large

_energy

splitting _(E~ -Eo)

moderate values of fJ

yield

a

satisfactory

result for the

groundstate

(0).

The

orthogonalization technique originally proposed by

Sorella et al.

[9] helped

_us to stabilize the

algorithm

^thus

avoiding

_severe difficulties

conceming

the well-known

« minus-

sign

_»

problem

for fermionic

systems.

^The

many-particle problem

is _now transformed into an

effective

one-particle problem.

First _we

approximate

the operator exp

(- p

JC

) according

to the Trotter-Suzuki

decomposition [7, 12]

e~ ^~~ _-

fl

_e ² ^° ^e~ ^~~~ _e ² ^°

,

~~

^~~

^~~ (4)

where JCo ^denotes ^the ^kinetic ^and

JC~

the Hubbard interaction term. The corrections _are of the order

e2,

_m denotes the number of slices in

imaginary

time and _E

= fJ/m their size. The many

body

interaction is _now transformed into _a _sum _over Hubbard-Stratonovich fields. The relevant contributions can be summed up

conveniently by

^standard Monte Carlo

importance sampling

techniques.

(4)

We

employ

the discrete Hubbard-Stratonovich transformation invented

by

Hirsch

[13, 14]

to treat the Fermion interaction

part

~~

£&~ ^A

£

_«, +,

~ ^~ "

i

_~ ^~ _~ '

,

(5)

j«, ±11 with

N=zh,, h~=n~i+n;1, 4i=nil-nil, U>°,

,

and cosh (A

)

= exp e

(U( ), ^(«,)

are

time-dependent

stochastic

Ising

fields.

2

The

algorithm

has been checked

against

exact

diagonalization

results for small

clusters,

for further details _see references

[10, 11].

According

to

Yang [15]

the relevant order

parameter

^for ^the examination of the supercon-

ducting properties

in _a

system

^with ^fixed

particle

number is the reduced

two-particle density

matrix _or rather the

Cooper pair

correlation function

(CPCF)

x~(ii)

₌

joy c)~ji cl-ii c~+i+jic~+i-ji ^io> ⁽⁶⁾

x ^measures the CPCF between _a

Cooper pair

of extension _m at site I and _a second _one _at

distance

f. Apart

from usual

in-plane

correlation functions

ill]

like extended _s-wave symmetry we consider

Cooper pair

structures with nodeless d-wave symmetry

(Fig. I),

x

(f )

=

( _z _lo At _{(I ) J(I}

+

I joy

,

(7)

with the _«

Cooper pair

creator »

A~

(I

₌

c)

_it

c)

~j

(8)

Hereby

one

charge

carrier is situated in

layer

I and the second in the other

layer 2,

_e.g. the

Cooper pair

_spans _over the two

layers

as shown in

figure1.

' ' ' ' ' '

--+---M--W-

' ' ' ' ' '

-+---M--W-

' ' ' ' ' '

-+---M--W-

' ' ' ' ' '

-+---,---M--W-

' ' ' ' ' '

-'---~--M--W-

-~--"--'--'-M~

' ' ' ' Layer 2

6t

' ' ' ' '

-+--- ----M--W

' ' ' ' '

-+--- -M--W

' ' ' ' '

--~--- -M--W

' ' ' ' ' '

--~--- M--W

' ' ' ' '

-~--+----~-- -W-

~'--"--'-~'-M'

^'

' ' ' ' L&Ye~1

~

Fig.

1.

Drawing

of the double layer Hubbard model with _a schematic definition of the nodeless d-

wave

pairing

correlation symmetry ; the

Cooper

pair _spans over the _two

layers

with _one

charge

carrier in each

layer,

the

position

of the fermionic creation and annihilation operators ⁱⁿ ^the

two-layers

system ^is

indicated. The

Cooper pair

distance is denoted

by

^I.

(5)

1046 JOURNAL DE PHYSIQUE I N° 4

To exclude the effects of any residual

quasiparticle-interaction

we focus _our interest _on the

vertex CPCF

by subtracting

the

corresponding one-particle

contributions. This leads to a

measure for the effective interaction between the carriers within the

Cooper pairs.

k ~~"~~(l

)

~ km

(1~ (°

_C~,

II Ci ₊ f, II

°) (°

^~,

21 Ci

+f, 21

°) (9)

A

macroscopic quantum

state is indicated

by off-diagonal long

range order

(ODLRO) [15],

which is

present,

if X~~"~~

approaches

a finite constant value («

plateau

value

»)

for

large cooper pair

distances

f.

xie"ex(f)

= cm + exp

,

(lo)

x

ie"x (f ) W

cm

(

i

1)

Kohn and

Sherrington [16]

have shown that the presence of ODLRO

implies

the existence of

a Meissner effect and thus

superconductivity.

Results.

The simulations _were carried _out _on _a Hubbard model up to a

system

^size 8 x 8 _x 2 with U

= 8 t and 0 _~ ti « t. We considered the _case of 0.15

charge

carriers per site. There has been

no severe minus

sign problem.

Figure

2 shows in _a

semilogarithmic (Fig. 2a)

as well _as in _a linear

(Fig. 2b) plot

our vertex correlation

X~~"~(i)

_versus _distance

I

_for _ti _=0.I

t. After _an

exponential decay

of

X"(i)

for small distances I the vertex reaches

obviously

the mentioned

plateau

value c~ ~a 10~ ^~ This

provides

clear evidence for

superconductivity.

With

increasing

the

interplane hopping parameter

to ti = 0.6 t the

plateau

value of the _vertex cPcF decreases

by

_one order

o.ooi

x"r'~(l)

m m

o.oooi

le-05

1e-06

2 3 4 5 6

Cooper

Pair Distance I

Fig. 2a.-Semilogarithmic plot

of the nodeless d-wave _vertex correlation function _as defied in equation (9) versus

Cooper pair

distance f _in the double

layer

Hubbard system. Notice the exponential decay within three lattice

spacings

and the long range finite

plateau

value as an indication for ODLRO.

Weak, but finite interlayer

hopping

parameter t~ = 0.I t, system size 8 _x 8 _x 2, other parameters as

described in the text.

(6)

o.ooi

0.0009

x"*~'"(l)

m

0.0008 0.0007 ^"

o.ooo6

~ o.ooos

0.0004

0.0003 ^"

0.0002

0.0001 ^"

~,m

^. _. ^. m

0

2 3 4 5 6

Cooper

Pair Distance I

Fig.

2b. Same system as

figure

2a in linear

plotting

8 _x 8 x 2 system, t~ = 0.I t.

of

magnitude (Fig. 3).

In addition for ti ₌ ^1.0 t

(I.e. isotropic hopping probability

within and between the

layers)

we found _no attractive interaction. In

figures 4a,

b _we

plot x~~"~

_versus

Cooper pair

distance

I

_for _ti

= 1.0 _t. The vertex CPCF shows

negative

values and converges for

large

values

off

_to _zero, _To

emphasize

this _we show the _same data in

semilogarithmic

and

linear

scaling.

Thus there is _an effective

short-ranged repulsion

in contrast to the above mentioned

long-

range correlation for ti = 0.I t.

o.ooi

x""'(')

_.

o.oooi

m .

1~05 .

~

"

le-UfJ

I j 4 5 6

Cooper

Pair Distance

Fig.

3. -Nodeless d-wave vertex similar to

figures

2a, b in _an 8 _x 8 _x 2 system ⁱⁿ ^this

figure

the

interlayer hopping

parameter ^has ^been ^increased to t~ = 0.6t. As _a result _we get a

plateau

value c~ which is about _one

magnitude

smaller than in the t~ ₌ 0.I t-case of

figure

2

indicating

_a _strong decrease of the

long-range

correlation value witli

increasing interlayer hopping.

Notice also the

negative

value for t~ ₌ ^I-o t

(increasing hopping)

_as shown in

figures

4a, b.

(7)

1048 JOURNAL DE

PHYSIQUE

I N° 4

o.ooi

o.oooi

m

l~os

1e-06

I 2 3 4 5 6

Cooper

Pair Distance

Fig.

4a.

-Decay

of the nodeless d-wave vertex correlation function witli tile

Cooper pair

distance f in _an 8 _x 8 _x

2-system

with t~ = I-o t, further parameters as in the text,

semilogaritlimic plotting.

Notice the negative values of tile CPCF

indicating

_a

repulsive

interaction.

Obviously

the correlation values do not reach _a finite

plateau

value for large

Cooper pair

distances and thus _no ODLRO is found in this system.

0

, m

,

mm

-5e-05 ^"

o.oooi

-o.ooois

-0.0002

-0.00025

-0.0003

0

oper air 1

Fig.

4b. Same system as

figure

4a in linear

plotting

8 _x 8 _x 2 system, t~ = 1.0 _t.

We also considered the extended _s-wave

pairing

channel. With the described set of

parameters

^there ^is no attractive interaction found. Smaller systems

give

similar results for the nodeless d-wave _as well _as for the extended _s-wave channel.

For the _case of15 §b

doping

_away from

half-filling,

I-e. 85 §b

particles,

_we

only performed

simulations for 4 _x 4 _x 2 lattices. For this small

system

^size we are unable to comment on

long

range behaviour of the correlation functions. But _we would like _to report a

positive (I.e.

attractive)

cumulated vertex CPCF for the nodeless d-wave _as well _as the extended _s-wave

channel,

the latter in contrast to the _case of 0.15

particles

_per site.

(8)

In the

thermodynamic

limit it

might

be

possible

to

give

_an

approximate

relation between the

plateau

value c~ and the energy gap of the BCS

theory allowing

_a direct calculation of the critical

temperature.

^For this estimation it is necessary ^to

extrapolate

from finite systems ^and therefore _to simulate _more and

larger systems.

At the

present stage

we are not able _to

give precise

results for the transition temperature. ^Further calculations with the

projector

method _as well _as with Hirsch's finite temperature

algorithm [17]

_are

planned

to support our simulations.

A pretty

rough

estimation

using

the above mentioned formalism for the

purely

electronic

system

seems to

give T~-values

which _are too low _to describe the copper oxide

superconduc-

tors,

especially

the Bi- and

Tl-compounds

with _more than loo K transition temperature.

To

explain

the

high

values of T~ in these materials we would

suggest

an additional _« booster mechanism _»

perhaps

ⁱⁿ ^the form of

phononic degrees

of freedom

[18].

In order _to check whether the nodeless d-wave channel is sensitive to

phonons,

_we

introduced _an additional

coupling

to local

phonons

between the

planes,

_as described in references

[4, 5].

Each of them

coupled

to four

neighbouring

sites in both

planes (Fig. 5,

«

plaquette

model

»).

^As ^shown ⁱⁿ

figure

6 there is _a

strong

^increase in the nodeless d-wave vertex CPCF with

increasing electron-phonon-coupling strength.

:....:....:... :.... :.... :... :...

: .lay«~

/ ~ / i / ~ /

Ill. Ill. Ill. Ill.

Fig. 5. - _Schematic ross-section

indicating

the

way of electron-phonon coupling. The filled circles

indicate

the

electronic

sites and

the

ph's

represent

tile _hononic degrees of reedombetween the layers.

Anharrnonic phonons are _coupled locally _in a _laquette model to _both

o.ooi

0.0008

0.ooo6

11.0004

X$(Z"

₊

11.l10112

0 _11,I 11.2 11.ll 0.4 0. 5 0.6 0. 7 0.8 0.9

El-Ph-Coupliiig Strength

_g

Fig.

6. Phonon-enhanced double

layer

_system _: Cumulated nodeless d-wave vertex correlation (I.e.

cumulated _over all distances

f)

for

a 4 _x 4 _x 2 system with

doping

8 15 fb, _t~

= 0.6 t as a function of the

electron-phonon coupling strength

g.

(9)

1050 JOURNAL DE

PHYSIQUE

I N° 4

Conclusions.

In conclusion our calculations

provide

clear numerical evidence for

superconductivity

in _a

purely

electronic system, ^the ^double

layer

^Hubbard ^model. ^While ^the

single layer

system ^is not

superconducting,

_our results

suggest

^that a weak but finite

interlayer coupling

leads to

superconductivity

in agreement ^with recent work of P. W. Anderson

[2, 20].

To reach the very

high

critical

temperatures

^of ^the oxide

compounds

it _seems

appropriate

to take _an additional

« booster mechanism _» into account,

possibly

also in the form of the above mentioned _« apex oxygen » oscillations. A second

T~-enhancing

mechanism

might

be the _van Hove-scenario

suggested by

Newns et al.

[19].

The actual nature of the

pairing

wave function _seems to be determined not

by

the basic

interlayer

mechanism but

by

these «residual interactions» caused

by phonons

_or other

sources.

Acknowledgments.

We

especially

thank P. W. Anderson for very

helpful

discussions. Moreover _we would like to thank K. A.

Miiller,

D. M.

Newns,

P. C.

Pattnaik,

U.

Krey,

J. Keller and K. F. Renk for their

help

and encouragement. ^We are indebted _to M.

Frick,

W. _von der Linden and H. de Raedt for their support ⁱⁿ

developing

essential parts ^{of the}

algorithm.

One of _us

(I.M.) acknowledges

the

hospitality

of the IBM T. J. Watson Research Center and the

Aspen

^Center for

Physics

where parts ^of ^this ^work were

accomplished.

Most of the

present

calculations have been

performed

_on

the CRAY-YMP and Intel

Hypercube iPSC/860

_at the German

Supercomputing

Center HLRZ

Jiilich _; the generous grant ^of ^CPU ^time ^is

acknowledged.

This work _was

partially supported by

«

Bayrisches Hochschul-Verbundprojekt

FORSUPRA _».

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