Flux-quantization in one-dimensional copper-oxide models

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Flux-quantization in one-dimensional copper-oxide models

C. Varma, A. Sudbø

To cite this version:

C. Varma, A. Sudbø. Flux-quantization in one-dimensional copper-oxide models. Journal de Physique

I, EDP Sciences, 1993, 3 (2), pp.585-589. �10.1051/jp1:1993125�. �jpa-00246744�

Classification

Physics

Abstracts

05.30 74.70V 71.30

Flux-quantization in one-dimensional copper-oxide models

C. M. Varma and A.

Sudb#

AT&T Bell

Laboratories,

Murray

Hill,

New

Jersey

07974, U.S.A.

(Received

19 October 1992,

accepted

26 October

1992)

Abstract. A calculation of the ground state energy

Eo(4l)

_as

a function of flux 4l in

a

one-dimensional'copper-oxide

model' _on

rings

of finite circumference L,

including

on-site inter- actions, and _{a nearest}

neighbor

interaction V, is

presented.

For V of the order of the

charge-

transfer _{gap or}

larger,

the model

extrapolated

to

large

L exhibits

flux-quantization

with

charge

2e. The

extrapolated superfluid

stiffness _appears,

however,

finite only for not too

large

V. These results suggest _a

superconductive

state at V of order the

charge-transfer

_gap of the

model,

but

a

paired

and

phase-separated

state at

larger

V.

Three distinct electronic models have been

proposed

for the

copper-oxide

based metals with

high-temperature superconductivity

and associated

phenomena.

Anderson

suggested

that the essential

physical

features of the materials _can be modelled

by

the twc-dimensional

(2D)

_one-

band Hubbard model

[Ii. Varma,

Schmitt-Rink and Abrahams

(VSA) presented

_a 2D model

consisting

of three orbitals _per unit cell

[2], namely

the Cu

3d~2-y2,

^and oxygen

2p~, 2py

^with

hopping

between

them,

_a

repulsive

interaction _on Cu

sites, Ud,

^and _{on oxygen}

sites, Up,

as well

as a

nearest-neighbor

copper-oxygen interaction V.

Independently, Emery presented

_a three- band model

[3],

^{in which the}

parameter

^V was assumed to

play

_no role. In reference

[2],

^it was

argued

that the three-band model with V included

(hereafter

denoted

as the VSA

model)

is the minimum model _necessary to characterize the essential

properties

of the materials. In

partic- ular,

the

parameter

V _was shown to be

important

in

inducing low-energy charge-fluctuations

in the metallic state. It _was also

argued

that such fluctuations

promote

_a

superconducting instability

and _more

recently

that

they

_may also lead to the anomalous normal metallic-state

properties

of the

high-Tc cuprates [4],

which at

1/2-filling

_are

charge-transfer

insulators. Here

we show that the

parameter

V is central in

inducing pairing

at T

= 0 in the one-dimensional

(ID)

version of the three-band model.

The ID version of the VSA model _may be written _as H ₌

Ho

₊

Hu

+

Hv,

where

Ho

_"

T(16)

+ A

~j _{[np,; nd,I],}

;

Hu

₌

Ud~j

nd,;~ind~;~i ⁺

Up ~j

np~;,inp~;~i,

(1)

; ;

586 JOURNAL DE

PHYSIQUE

I N°2

Hv

₌

V~nd,I[np,;-i+np,;).

I

Here,

_{na,; =}

£~

_n~~;~a

=

£~ cl

; ~ca~;~a, where

~Y denotes _an orbital index

(~Y E

(Cu, O)),

and

a a

spin

index

(a

E

(i,1). Thr1~lghout, £;

is taken to run over

unit-cells,

chosen such that the _oxygen is to the

right

of _copper. We have introduced the

parameter

A _e

(Ep Ed)/2,

with

Ep

>

Ed

in _a hole

notation,

where

Ep

^and

Ed

_are site

energies

_on the _oxygen and copper

sites, respectively,

and the _zero of energy has been chosen at

(Ep Ed)/2. Furthermore,

the

quantity T(16)

is the kinetic _energy

operator

^{of the}

system

in the _presence of _an external flux

4l,

and will be

specified

below.

The model of

equation (I)

with

T(16)

= 0 is

exactly

solvable for

arbitrary

L in ID

by

^the

transfer-matrix method

[5].

^For Ud

large compared

to

Up

^and

A,

and

density

_away from

1/2-filling, increasing

V to

tl(A)

leads to _a combined

charge-transfer

and

phase-separation instability.

In this

instability,

the average

charge

_on the

p-orbitals

increases at the _expense of the

charge

_on d-orbitals. The former _are either

nearly doubly occupied

_or

empty,

_as reflected

by

the

compressibility tending

to

infinity.

Mean-field calculations also show such instabilities with

finite

kinetic _energy [6], but in addition reveal _a

region

of

parameters

where _s-wave

superconductivity

exists without

being preempted by phase-separation.

A central

question

is whether _{one can see,}

through

exact

calculations,

that kinetic energy favours

superconductivity

over the

charge-transfer/phase-separation

instabilities for _any

filling

N and in _any

region

of the

parameter

_space

(Ud, Up,

V,

A),

with

purely repulsive

bare interactions.

To

explore

these

issues,

_we have considered the

ground

state energy

Eo(16)

of the Hamiltonian of

equation (I),

in _a

ring geometry

^with

L/2

number of Cu O unit

cells, (L being

_{an even}

integer representing

the total number of sites in the

problem).

The

ring

is threaded with _a

dimensionless Bohm-Aharanov flux 16

by applying

_a constant vector

potential

of

magnitude

A =

(hcle)16/L along

its circumference. The effect of the vector

potential

A

=

(hcle)4l/L

is included

by

_{a gauge} tranformation

(taking

h _{= e = c =}

I),

_{ca,m,a - c~~m~a}

exp(im4l/L).

Thus,

_we have

T(4l)

₌

-tpd ~

(e~~/~ cl

; ~cp,;,a ⁺

e~"/~c(

; ~cp,;-i,a ⁺

h.cj ⁽²⁾

I,a

From the

ground

state energy

Eo(4l)

_one obtains information about

flux-quantization

and

possible superconductivity:

In _a normal one-dimensional

ring,

the

ground

state energy is _a

periodic,

even function of 4l with

period

_{4lo +}

hcle.

If the flux 4l is measured in units of the

flux-quantum 4lo,

_we then have

Eo(4l)

₌

Eo(4l

+

n);

_{n =}

0,+1,+2,.. Hence,

for _a normal metallic

phase,

_one then has stable

phases

at

particular

values of the

flux,

4l

=

0,+1,+2.., leading

to

flux-quantization

in units of

hcle [7].

In _a

superconducting

state _one further has

Eo(4l)

₌

Eo(4l

+

n/2)

in the

thermodynamic

limit L _{- oo,}

showing

that _new stable

phases

appear also at 4l

=

+1/2, +3/2,.,

in addition to the _ones at 4l

=

0, +1,+2,..

in the normal metallic state. In

systems exhibiting superconductivity

flux is thus

quantized

in units of

4lo /2

=

hc/2e [7],

^{hereafter referred} to _as anomalous

fluz-quantization.

Recently,

the

ground

state energy

Eo(4l)

for the

repulsive

and attractive one-band Hubbard

ring

with finite circumference L and onsite interactions

only,

_was considered in detail

[8].

^This

model is

exactly integrable

for

arbitrary

flux

4l,

and _exact results for

ground

state

properties

are thus available for

arbitrary

L. For the attractive _case, which is known to exhibit

singlet

superconductivity,

it _was found that the results for this

particular

_{case may} be written

For the attractive

Hubbard-model, f

is _a

length

associated with _{a gap} of

spin-excitations

in the

system,

A ₌

A(U, N)

is _a coefficient which

depends

_on details of the

model,

U is the onsite

interaction,

and N is the number of

particles

_on the

ring.

Another

quantity

of

interest,

which should be considered in

conjunction

with

Eo(4l

+

n/2) Eo(4l)

for the _purpose of

detecting

superconductivty,

is the

superfluid

stiffness

02[Eo(4~)/Li

(4) PS(~)

_"

d[4~/Lj2 '~"°'

This

quantity

should

approach

_a finite _{constant as} L _{- oo} in _a

superconducting

_or

perfectly conducting

metallic state.

Equations (3)

and

(4)

_are useful

diagnostic

tools when

looking

for

superconductivity

in the VSA model

equation (I),

with A ₌

A(Up, Ud,

_V,

A, N).

We _use

Lanczos-diagonalization

to obtain the T ₌ 0

eigenvalues

and

eigenvectors

of the Hamiltonian

equation (I).

In

considering

numerical

approaches

of this

kind,

which

by necessity

are limited in

system

size

L,

one should bear in mind that states other than

superconducting

ones may

produce

anomalous

flux-quantization.

For

instance,

it _was

recently

noted that

peri- odicity

in

Eo(4l),

with

periodicity 4lo/2,

is found in exact

diagonalization

studies of small

rings

of the

repulsive

one-band Hubbard model

[9].

We

have, however,

verified that _no anomalous

fluz-quantization

_occurs in this model when states of _a

given

total

spin

S _are considered for all 4l. The

crossing

in _energy of _states of different

spin

^S occurs due to

degeneracies

in finite

systems,

^{and associated} Hund's rule effects which _are

negligible

in the

thermodynamic

limit L - oo. Paired states which _are

crystalline (charge density waves)

_or

phase-separated

will also have _a

tendency

to

4lo/2-quantization,

but in these _cases ps will decrease with

L,

whereas in

a

superconducting

_or

perfectly conducting

metallic state ps should

approach

_a

finite

constant

as L

- oo.

Hence,

to test for

superconductivity,

_one must calculate both

Eo(4l)

and ps, and consider their behavior with

increasing

L. In _our

computations,

_we have been limited in

system

sizes L < 12 sites for

fillings

N _>

L/2

of

interest,

I.e.

systems doped beyond half-filling-

We _now turn to the details of the calculations with the model of

equation (I).

The choice of the

parameters

is made such that at

half-filling,

the model is _a

charge-transfer

insulator

with _a gap of about 2

(in

units of

tpd),

with

80$l

_{or more} of the

charge residing

_on the d- orbitals

(corresponding

almost to _a

Cu++

_{state in the Cu O}

materials).

With

Ud

_*

6,

this is ensured for both A and

V,

in _a range between 0 and 4. In this range of

parameters,

as

V

increases,

_one obtains _a

rapid

transfer of

charge

from

copper-orbitals

to

oxygen-orbitals for systems doped beyond half-filling,

at the few

fillings

_{we can} consider. This _was also _seen in

mean field calculations

[6].

^{It is this}

region

of

parameters bordering

_on

phase-separation

which is the

region

of interest to us when

searching

for

superconducting ground

states.

Figure

I shows

Eo(4l)

obtained in the VSA-model with

parameters

L ₌

10,

^N ₌

8, tpd

_"

I,

Ud

_"

10, Up

=

0,

and 2A ₊

Ep Ed

_" ² for three values V ₌

1.0,

V ₌ 3.0 and V ₌ 4.0.

For the latter two

values,

the

system

^exhibits anomalous

flux-quantization,

whereas it does

not for V

= 1.0. A critical valve

of

V is

therefore

needed to _see

pairing.

The

filling

in this

case

corresponds

to

60$l doping beyond 1/2-filling.

For small V <

I, Eo(4l)

is maximum at 4l =

1/2,

and has

period I,

characteristic of _a normal metallic state. As V is

increased,

_a

local minimum

develops

at 4l ₌

1/2

_{as a} result of level

crossing

between states of different

total momenta k ₌

2~m/L;

_m

=

0,+1,+2,...

It is found that

Eo(1/2) Eo(0)

^decreases

rapidly

_as V is

increased,

consistent with

equation (3):

As V increases the effective attraction

on oxygen sites increases and

produces

_more

tightly

bound

pairs _[10].

Note that _we

expect

that the

pairing

induced

by

V will be _more

pronounced

when the

dimensionality

D _> I: the coordination number of the _copper sites in

Hv, equation (I),

increases _as D increases. The critical value of V necessary ^to induce

pairing

will thus decrease when D increases. As the

588 JOURNAL DE

PHYSIQUE

I N°2

L=I$ N=8 ^O V

= I.O

0.05

Ud"10,U~=0

^{~ ~/"3.°}

A=2.0 ~/"4.0

j~

jJ j

j ^'

, j j

,

j O

i

. '

, o

.

, uJ

~

J

=

'

J

/

~

',

i

'~

/ l

J

u~

', ~__-~ _{A/ i}

Q

i,

~, J

j

/ _r"

x 0 125

',

j

p

Q ,

J /

',

, J

,~

_, /

J O

,,

'4~-O~~

J,'

° °~

'UT'

o o.5 1.o

if 2n

Fig.I.

Ground state energy

Eo(4)

_us. flux 4 for the 1D VSA model with L

= 10, N

= 8, Up = 0, Ud = 10, 2a

= 2, and tpd ₌ ^{for various values of} V. Note the

tendency

to

halfing

of the

periodicity

of

Eo(4),

characteristic of

flux-quantization

with

charge

2e, when V

= 3.0 and V

= 4.0 but not at

V = 1.0. Note also the

rapid

decrease of ED

(1/2)

ED

(0)

_as V increases, consistent With _a reduction of the

Cooper-pair

size v4th

increasing

V. The thin broken lines _are

guides

to the _eye.

onsite Coulomb

repulsion

on oxygen sites Up ^is

increased,

the effective attraction _{on oxygen} due to V is reduced.

Pairing

is

consequently suppressed,

and is not

present

^when

Up

m

Ud.

The I0-site chain with N

= 8 shows anomalous

flux-quantization

for _a

large

_range of V.

Furthermore, by inspecting

the

dispersion

of the

ground

state energy

Eo(k,

4l

=

0),

it is _seen that the

system

^maintains

quasi-particles

with _mass within _a factor of 2

(I.e. Eo(k,4l

₌

0)

remains

dispersive)

from that at V ₌ 0 in the entire range of 2 < V < 4 _we have

considered,

and where

4lo/2-quantization

is observed. These _are

encouraging signs

of _a

superconducting ground

state.

However,

the 10-site chain is not convenient for

carrying

out the finite-size

scaling analysis

mentioned

previously.

We therefore next consider _a 6-site and _a 12-site

chain,

which also allows _two

systems

with identical

fillings

to be

investigated.

For the 6-site chain _we have used N ₌

4,

^while for the 12-site

chain,

_we have used N ₌

8,

which

corresponds

to

33$l doping beyond

half

filling.

The

filling

is lower than in the _case of the I0-site

chain;

observation of anomalous

flux-quantization

thus

requires larger

values of V.

For

Ud

_"

6.67, _Up

₌

0,

2A ₌

1.33,

and V

=

2.67,

_we have

AE~(L

=

6)/AE~(L

₌

12)

₌ 0.3 and

ps(L

₌

12)/ps(L

₌

6)

₌

0.67,

whereas for V

=

4,

the former is 0.07 while the latter is 0.25.

Here,

_we have defined

1hE~

_e

Eo(16

=

~) _Eo(16

₌

0).

For _a

superconductive ground

state and for L _»

f,

_ps ^should scale to a constant while

AE~

should decrease

exponentially

with

large

L. For V ₌ 2.67 _we find

using equation (2)

that

f

_m

6,

so we are not in the

asymptotic

regime

and the small decrease of _ps with

increasing

L is consistent with _a

superconducting

state.

However,

for V

=

4,

we have

f

m

2;

in this _case the

sharp

decrease of _ps with L is

suggestive

of _a

paired

but

non-superconducting

state.

Furthermore,

from

Eo(k,16

=

0)

^for

L ₌

12,

N ₌

8,

the lowest _energy state is _seen to be

essentially dispersionless,

consistent with _a

pinned

CDW _{or a}

paired/phase-separated state,

^{the latter also}

previously

indicated

by

mean-field calculations

[6].

^{In this}

context,

we note that for L ₌ 12 and V

2 4,

_we find that the chemical

potential

_{~ =}

Eo(N

+

I) Eo(N)

at N ₌ 8 is close to its value at

half-filling,

_a

signature

of

phase-separation.

This

suggests

a

superconducting

state for

parameters locating

the

system

close to _a

phase-separation instability.

The above results _were

presented by

_one of _us

(C.

M.

V.)

_on the occasion of the memorial

symposium honoring

R. Rammal.

They

_are

part

of _an

investigation

carried out with collabc- ration of T.

Giamarchi,

R. T.

Scalettar,

and E. B. Stechel.

They will, together

with further

details,

also be

published

elsewhere.

Acknowledgements.

I

(C.

M.

V.)

wish to thank Benoit Doucot for

giving

me the

opportunity

to pay

homage

to my friend Rammal. I

got

to know Rammal _very well

during

the _year he _was kind

enough

to

accept

my invitation to visit the Theoretical

Physics

Research

Group

at Bell Labs. Besides _a

deep

interest in all _manners of

science,

_we shared the

special sensitivity

and

perception

which _comes from

living

in _our

respective

societies _as outsiders.

Through

Rammal I learnt much about the

academic and scientific culture of France which he

deeply loved, although

in _a bitter-sweet fashion.

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(1987).

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Varma,

S.

Schmitt-Rink,

and E.

Abrahams,

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Varma, Phys.

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