Extended proof of long-range order in the two-dimensional quantum spin-1/2 XXZ-model at T=0

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Extended proof of long-range order in the

two-dimensional quantum spin-1/2 XXZ-model at T=0

Hans-Aloys Wischmann, Erwin Müller-Hartmann

To cite this version:

Hans-Aloys Wischmann, Erwin Müller-Hartmann. Extended proof of long-range order in the two-

dimensional quantum spin-1/2 XXZ-model at T=0. Journal de Physique I, EDP Sciences, 1991, 1

(5), pp.647-657. �10.1051/jp1:1991109�. �jpa-00246359�

Classification

Physics

Abstracis

75.10Jm

Extended proof of long-range order in the two-dimensional

quantum spin-1/2 XX2kmodel at T

=

0

Hans-Aloys

WischJnann and Ervin Mfiller-Hartmann

Institut ffir Theoretische

Physik,

Universitit _m K61n,

Zfilpicher

StraBe 77, D-W 5000 K61n 41,

Germany

(Received25

January J99J,

accepted

J

February

J99J)

Abs«act. We _prove the existence of

long-range

order

(LRO)

in the

ground

state of the 2D quantum

spin-1/2

XXZ-model for all

anisotropies

0 WA _« 0.22 and Am1.47. We achieve this extension of

previous proofs by developing

_an

improved

variant of the

Dyson-Lieb-Simon proof

of LRO in quantum spin systems and by deriving improved _upper and lower bounds for the

ground

state energy of the XXZ-model.

1. Introduction.

In recent years, there has been _a resurgence of interest in low-dimensional

quantum spin models,

which

considerably

increased with the

discovery

of

high-temperature superconductors ii

]. Due to the

layered

structure of these

materials,

most of the

early attempts

to describe the

novel

physics

_were based _on electronic and

magnetic

correlation effects in two-dimensional models. The

simplest

of

these,

the Hubbard and the

Heisenberg

models have since been the

object

of innumerable

studies,

and the

investigation

of the

ground

state

properties

of these

models has outlasted the somewhat naive

hopes

of _a

simple

electronic mechanism for

superconductivity.

The most

interesting

and most

extensively

studied

property

of the

ground

state of the

antiferromagnetic Heisenberg

model is the

decay

of the

spin

correlation functions with distance.

Although

much numerical evidence for the existence of _a

Nkel-type antiferromagne-

tic

long-range

order

(LRO)

has been

accumulated,

_a

rigorous proof

for the square lattice is still

missing.

On the other

hand,

there has been

significant

_progress in

higher

dimensions and for the

anisotropic generalization,

the XXZ-model _:

H

=

£ (S) )+

+

St _)Y

₊

AS) )f)

with A _m 0

,

(1)

(<Jl

where the

exchange

interaction J has been set to

unity.

A number of authors

[3-6]

extended

the scope of the

original proof by Dyson,

Lieb and Simon

[2]

for the

Heisenberg

antiferromagnet

and

generalized

it to the XXZ-model. In this _manner, Kubo and Kishi

[8]

were able to show the existence of LRO in three and _more dimensions for all

spins

S in the whole

antiferromagnetic region,

in two dimensions for SW I also for all

Am

0,

_as well _as for S

=

1/2

in the KY-like

region

0 WA

~ 0.13 and the

Ising-like region

A _m 1.78.

Ozeki,

Nishimori and Tomita [9] ^{extended this}

proof

to A

m 1.72. Their

proof

for 0 _« A _<

0.20, however,

is not

rigorous

: it _uses «variational _»

energies

which Suzuki and

Miyashita [13]

^calculated in _a very

good

but uncontrolled

approximation

as upper bounds for the _energy. The _same

energies

_are also used in the

proof

for 0 _« A _« 0.59

by

Nishimori and

Ozeki

[10].

In

addition,

this last result _as well _as the

proof

for A

m 1.10 also contained in

[10],

is based upon the

assumption

that the

nearest-neighbour

correlation functions _are monotonic functions of the

system

size. This

assumption

_was verified

numerically

in

[10]

for the

10, 16,

18 and 20

spin

_systems with Oitmaa-Betts

[12] boundary conditions,

but _we have found that the

x-component

^{of these} correlations does _not show _any

monotonicity

for A=0.8 and A

=

0.9 when _one includes the 26

spin

_system

('). Although

this _range of A is outside the

region

of interest for the

proof,

a doubt is _{cast on} the

validity

of the

assumption,

since

larger

systems may exhibit this kind of behaviour _at smaller values of A. On the other

hand,

the

proof

for the

Ising-like region

A

m 1.67

given

in

[10]

is

rigorous,

_so that the existence of LRO in the

ground

state of the XXZ-model has been shown for all

dimension,

all

spins

and all

anisotropies

with the

exception

^of

S=1/2

_on the _square lattice for the

anisotropies

0.13 _w A _w 1.67.

In this work _we present an extension of the above

proofs

to the KY-like

region

0 _w A _w 0.22 and to the

Ising-like region

A _m

1.47,

thus

closing

_as much of the

remaining

_gap

as is

possible

with this method. The

improvement

is achieved

by optimizations

on three

aspects

^{of the}

existing proofs

in section

2,

_we

present

an

improved

variant of the

Dyson-

Lieb-Simon

[2] (DLS) proof,

in section 3 _we derive

improved

lower bounds for the _x-

component

^{of the}

nearest-neighbour spin

correlation

functions,

while section 4 is devoted to

improved

variational

energies. Finally,

these parts are assembled and the results _are summarized in section 5.

2. The method.

Following

Kubo and Kishi

[8],

_we define the Fourier transform of the I-th

component

^{of the}

spins 6j

_{on a} D-dimensional

hypercubic

lattice A

(with

its

reciprocal

lattice

A*)

and the

positive

semidefinite

quantity gl'~

₌

(S[ S[~)

^{that satisfies the}

^following

sum rules _:

( _{z (Sk Sk)} =( _{z gl'~}

_and

₍₃₎

ReA peA.

( _{£ (6lt Sk}

+

e~)

"

j _{£ (gl'~}

C°S

pm) (4)

Re A pe A"

(1) Our numerical deternfination of the ground state correlation functions

produced

the

following

values for the 18, 20 and 26

spin

systems:

(6~S~)~~=-0.13044,

-0,13040, -0,13044 and

$~ S~j

~ ~ ⁼

0,12477, 0.12474, 0.12501

In the

thermodynamic limit,

the correlation functions become

independent

of the lattice sites R and the bond directions _e~, and _we obtain

(S( S()

=

i~~~

~ g)~~ ⁺ lim

( _g(~

_and

₍₅₎

Bz

(2

_gr

)

_A

- m

($ _S(

=

~~~

~ gj~~

j ⁽

cos p~ + lim

( _g(~

,

(6)

Bz

(2 ")

m=i Al -m

provided

the

gl'~

are bounded from above for all _p #

Q. Here, Q

₌

(gr,

_gr,..

)

is the _wave vector

corresponding

to

antiferromagnetic LRO,

and the second term _on the

right-hand

sides of

(5)

and

(6) is, just

the related order

parameter

lim

g(~

₌ ^lim

~

£ exp(-iQR) Sk I= ^{(ml'~)~ (7)}

A

- m

~

A

- m

A

ReA

~

A

rigorous proof

of LRO _can thus be obtained

by deriving

_a lower bound for the left-hand side and _a smaller upper bound for the

integral

_on the

right-hand

side of either of these

equations.

At this

point

_we remark that _we do not have to choose between

(5)

and

(6),

but _can instead

investigate

the linear combination

IS Sii

+ «

<Si Sol ~z

~/)~ ^~i~ _la i _{~l C°S}

Pm ⁺

+

(a

₊

I)

lim

g(~, (8)

(Al -m

A(

which reduces to

(6)

for _a

= 0 and to

(5)

for infinite _a. In

tiis _variant, rigorous proofs

_can be obtained for all values _{a m}

I,

but

negative

values do not

yield improved results,

so that _we shall

only

be concemed with _a m0 in the

following.

The introduction of the

parameter

a thus leads to a

family

of

proofs containing

the best

proof

conceivable

using only

site-

diagonal

and

nearest-neighbour

correlation functions.

The

required

_upper bound for

gl')

_was

originally

derived

by Dyson,

Lieb and Simon

[2]

for the

Heisenberg

and the KY-model in three and _more dimensions _at finite

temperatures.

Neves and Perez _[3] then showed that the two-dimensional

problem

_can be treated in the

same manner, as

long

_as the

zero-temperature

limit is taken before the

thermodynamic limit,

while

Kennedy,

Lieb and

Shastry [4] directly

derived the _same bounds in the 2D

ground

state.

These results _were

subsequently generalized

to the

anisotropic

_case

by

Nishimori _et al.

[7],

who obtained the

following

formulas for _zero

temperature

in the

thermodynamic

limit

g(I)

_~

j(I)

^I

₍ B(I)

c (I))1<2 j~ ^{(x), ~)} ~~

~,~-

ⁱ j~ ^(z) ~~ ^{~ ~,}

~-

ⁱ ~~~

P P

~2

P P ' P P ' P P '

D D

Cj~)~ ^~~ =

2

£ (1

A _cos

p~) lxx)

~

2

£ (A

_cos

p~) (zz)

~ ,

C)~)

₌ 4

lxx)

~

E~

,

m i m =1

(lo)

D D

Ej

=

£ (I+cosp~), E~

₌

£ (I-cosp~), (11)

m=i m=1

where the

following

abbreviations for the

nearest-neighbour spin

correlation functions have

been introduced :

(xx)~

₌

(S(Stj(A)

and

(zz)~

=

($S()(A). (12)

Inserting

the above

expressions

for

j)~)

^{for the} two-dimensional _case into

(8),

_we obtain the

following

sufficient condition for LRO in the

z-component

^{in the}

Ising-like region

A _{m :}

~

j- ^lxx)

_~

~~~~4

~

4 ^" 2 A

'~~~"~

_~~~~

with

where the

«

+

» indicates

that

the

integration is estricted to _that part _of

which the _ntegrand

is

_positive. imilarly, the ollowing ufficient condition

for

RO in the

jxxj~+f> jxx)~.fi.r~(«). ₍₁₅₎

Here _we have used the fact that

~

j2-cospi ^{-~°~P2. (2 a)}

⁺

~~~"'~~

"

i[z 2('1)~

_~

~ ~°~~~

~l~~[lospi

xC°~P~.

(-

_cospi

°SP2~)

~~~~

+ ~ _{~ cos} pi + COS p2

il

is _an

increasing

function of _{x, so} that _{we were} allowed to

replace h~(a,rj) by h~(a,

I

)

=

r~(a )

because the

argument jzz )

~ + A

lxx)

4

(1?)

ri "

j xxi

~ + A

jzzj

4

is bounded

by (rj

_w I. This _can

easily

be verified

using

the

equations (14a)

and

(14b)

of Nishimori _et al.

[7].

^These

equations

also _prove the

inequality (zz)~«- lxx)

~

for

0 WA _w I which has been used to

simplify

the

right-hand

side of

(15).

Following

Nishimori and Ozeki

[10],

_{we now} transform the above conditions

(13)

and

(15) using

the

monotonicity(2)

of the correlation functions

(xx)~m lxx)

for Am I and

lxx)

~ m

(xx)

~

for A _m 0 _as well _as variational values for the

ground

state

energies

_per bond _: e~

(A)

_W

e0(A)

#

2

jXX)

4 ~ A

l~~)

4'

(18)

We

finally

obtain the

following

sufficient conditions for LRO in the z-component ^{in the}

Ising-

like

region

A m I _:

~y

j- ^(XX)j.A

~v(4)"4~+2~~~)l~

~

°~2("), (l~)

~2)

_This

can

rigorously

be shown

using simple

variational arguments.

and for LRO in the x-component ^{in the} KY-like

region

0 _« A «1:

~

+ ^" »

fi _r~(

« for _{« »} o

(20)

fi

₄

_-(xx)~

2

3. Lower bounds on the

spin

correlation functions.

In order to be able to make _use of these

formulas,

we need

good

lower bounds _on the

correlation functions

lxx) (at

the

Heisenberg point)

_as well _as

lxx)

~

(at

the

KY-point),

that

can

evidently

be deduced from lower bounds _on the

ground

state

energies. Slightly generalizing

_an

elegant application

of the variational

principle

that _was

developed by

Nishimori and

dzeki _[10],

_these

can in turn be derived

by numerically calculating

the

ground

state

energies E#( ((y)

of the

inhomogeneous

Hamiltonian

HI

=

£ (y($~ f+

+

St _fY

+

AS) )f) (21)

(;y)

on a finite

plaquette

with open

boundary conditions,

where the

(j

^m0 represent ^bond-

dependant

interaction

strengths. Using

the

ground

state of the

homogeneous

model _on the infinite lattice _{as a} variational _{test state} and

taking advantage

of its translational and

rotational

symmetry,

we then obtain

(H/)

=

£ (~ (2 (xx)

~ + A

(zz ) ~)

m

E#( _{( (y ) ) (22)}

( ,j)

This leads to the desired lower bounds _on the

ground

state

energies

_per bond for the infinite

homogeneous

model

Et( jzj1 )

(23)

~°~~~

_~

z ij

^~

and

finally

to the

required

bounds _on the correlation functions _:

E(( (/y ) El ₍ (y) )

( xxi

~ ^m

and

( xxi

_m

(24)

2

£ (y

3

£ (j

The

remaining

task is to

judiciously

choose the set of interaction

strengths ((j)

_{so as} to maximize these bounds. Since _a

systematic study

of all

possible

sets is not feasible for

large plaquettes,

_we have used the

following

heuristic rule that has

always

_proven to

yield

_very

good

bounds: for the _case of _a

plaquette

of

Nx by N~ ^spins,

^the interaction

strength

(~

^{for the bond}

(ij)

is determined _as the function

S(x, Y)

=

Ii

^{+ C°S}

_1~ i~" _Ii

+ C°S

1~ i~" 1, ⁽²⁵⁾

evaluated at the centre of the

bond,

where the

origin

of the coordinate system ^{has been}

placed

at the centre of the

plaquette

and the bond

length

has been set to

unity.

We have been able _to determine the

ground

state

energies

of

HI

and

H/

with these

interaction

strengths

_on the 5 _x

4,

6 _x 4 and 6 _x 5

plaquettes using

_a modified Lanczos

method (3). The

resulting

lower bounds _on the correlation functions _{are :}

(xx)

_m 0. II

764, 0.11681,

0. II 524 and

(26)

(xx)~

_m

0.14094, 0.14034,

0.13942

(27)

respectively.

The

comparison

of the best obtained bounds with the

numerically

determined

lxx)

correlation functions for the

16, 18,

20 and 26

spins

systems

(using

Oitmaa-Betts

[12]

boundary conditions)

in

figure

I

clearly

shows that _our bound _on

lxx)

~

is excellent in the KY-

region

and

quite satisfactory

in the

Ising-like region.

4.

Upper

bounds _on the

ground

state

energies.

To

complete

the

proof,

_{we now} have to derive

good

_upper bounds _on the

ground

state

energies.

For this _{purpose, we} start from the Nkel-state

N)

that has

positive z-components

for the

spins

_on the A-sublattice and divide the infinite 2D lattice into

equal rectangular plaquettes P,.

We then

investigate

the

following

variational state

~P~)

^that was

originally proposed by

Bartkowski

ill]

and

subsequently

studied _on 2x2

plaquettes by Ozeki,

Nishimori and Tomita

[9]

(~P~)

=

fl fl' (I

₊

ASP S()(N), (28)

P, (<,b)eP,

where

fl'

is restricted to nearest

neighbour pairs jab)

with _as A and be

B,

and

A is _a real variational

parameter. Furthermore,

the conventions S+ =S~+iSY and

S~

=

S~- iSY will be used for the ladder

operators.

^Due to the fact that these

operators

commute

provided they

_{are on} different lattice

sites,

the calculation of the _energy is

04

20 18

°6 18

08

~ li

,,

~°

V ,>,e.3,4.5

iz

14

18

o 5 1-o 1.5

A

Fig,

I. Correlation functions

lxx)

~

for the 16, 18, 20 and 26 spin systems ^with Oitmaa-Betts [12]

boundary

conditions

(curves) compared

to the

rigorous

lower bounds

(xx)

~

and

(xxi (straight lines).

The inset shows the XY-like

region

_on an

expanded

scale.

(3) ^{For the 6} x 5

plaquette,

_one Lanczos step consumed

approximately

10 minutes CPU time _{on one} processor of _an

IBM-3090-6005/VF,

and 35 steps were necessary to achieve convergence.

straightforward

and leads to

ev

(A)

=

£ _{£ (If-}

+

AIl$)/Io

₊

£ _£

_'

_{(AII I])/Ii (29)}

ab)e P (<,b

Here M is the number of

spins

_per

plaquette

and

£'

^{is restricted} to nearest

neighbour pairs jab

with _{a e}

Pi

and b _e

(P~

U

P~),

where

P~

is the

right

and P~ ^{is the} upper

neighbour

of

plaquette Pi.

The introduced

expectation

values I _can be

expressed

_as

Io(A

=

(~P([ ~P() ⁽³⁰⁾

1](A )

=

(~P([S[[ ~P() ⁽³¹⁾

1(~_ (A

= ~P

([ ^St Sp

₊

Sp St

~P

() _{/2 (32)}

I~(A

=

(~P([ S( S([ ~P() ⁽³³⁾

with the

help

of

~P()

₌

fl' (I

+

ASP St ) [fir)

,

(34)

jab)eP

where

fir) represents

the

configuration

of the

spins

_on

plaquette

P when the infinite lattice is in the Nkel-state

)N).

As _we shall

show,

_an evaluation of these formulas leads to very

satisfactory

variational

energies

for the

Ising-like anisotropies

Am

I,

_even if the lattice is divided into _very small

plaquettes.

It thus appears

amazing

^{that this} test state is

seldomly

^used for the KY-like

anisotropies.

For 0 WA _w I it is of _course

advantageous

to renumber the

coordinates in the Hamiltonian

(I)

and

study

the

equivalent

form

ji

=

£ (AS/ f~

₊

Sf _fY

₊

S) f~) (35)

(;j)

=

£

^~

(St )+

₊

S/ ()

+

£

^{~ ~}

($+ §

+

Sj )+ )

+

S) )~ ⁽³⁶⁾

j,~~ 4

;~j 4

We have calculated the energy

i~ (A )

of the test state ~P~ with

respect

to

ji. _Noticing

_that

the first _sum in

(36)

does not contribute because it does not preserve the _z

component

^of the total

spin,

_we obtain _:

iv (A)

=

~j £ ((A

₊ I

I~t_ /2

+

Ijf)/Io

₊

£ _{£' (Ii If)/Ii (37)}

jabje P jab>

Denoting

the _sums of the

expectation

values I

by

j~_

£, ^~a~b

j~

£

~ab j~

£

^~ab

^(3~)

z z z, zz zz, + + ,

(<,b) (ab)_E P jab )EP

the variational

energies

_can

finally

be

expressed

_{as :}

e~ =

£ _{((T~_}

+

AT~~)/Io

+

(AT~)/I/)

^and

(39)

i~

=

(A

₊ I

) T~ /2

+

T~~)/Io

+

T~/1() ⁽⁴⁰⁾

we have written _a

computer

program ^to

symbolically

evaluate

Io

and the _sums T for the _case

of 2 _x

2,

3 _x

3,

4 _x 4 and 5 _x 5

plaquettes.

The details _can be found in

[14],

and the results for the 5 _x 5

plaquette

_are

given

in the

appendix.

The

subsequent

numerical minimization of the

energies

with

respect

to A for 201

equally spaced anisotropies

in the range 0 WA _w 2 then

provided

us with the

required

_upper bounds _on the

ground

state energy.

We _were able _to further

improve

these values

mainly

in the

vicinity

of the

Heisenberg point by investigating

the

following generalization

of the test state

*i)

"

fl C~P(~

_'l

(Nil Nil))' n' (I

₊

ASP St ) l~>

,

(41)

P, (ab)_eP,

where

Nl'/

_and

_{Nl'i~ represent}

_the number of

parallellantiparallel neighbouring spins

_on the

plaquette P;.

The

original

test state

corresponds

to

Y~ ⁼

0,

and

Y~ ⁼ 0.03 leads to

slightly

better variational

energies

for all

anisotropies

whereas

Y~ ⁼

0.06 leads to

slightly

_worse

values. Since the

improvement

is _so small that the scope of the

proof

is not

extended,

_we refrain from

writing

down the

corresponding

_sums T. The

energies

for

Y~ ⁼ 0.03

have, however,

been used in

determining

the bounds of table I and have been included in

figures

2 and 3 which present the variational

energies

for the KY-like and the

Ising-like regions.

It is evident from these

figures,

that _we have been able _to

considerably improve

the variational

energies

_over the whole range of

anisotropies.

Table I.

Limiting

^values

A~,~(a) for rigorous proofs of

LRO in the z-component ^and

A~~~(a) ^for rigorous proofs of

LRO in the _{x- as} well _as the y-component.

«

r~(«) A~~(«) A~~~(«)

0.0 0.646803 1.52 0.19

0.1 0.758695 1.49 0.21

0.2 0.877613 1.48 0.22

0.3 1.001576 1.47 0.21

0.4 1.129471 1.47 0.20

0.5 1.260509 1.48 0.19

0.6 1.394080 1.49 0.17

S. Conclusion and summary.

Assembling

the

improvements

that _were

presented

in the

preceding

three

sections,

_{we are}

now able to extend the scope of the

proof

of LRO. In section 3 _we derived _a lower bound of

lxx)

_m 0.11524 _on the

x-component

of the

nearest-neighbour spin

correlation

function,

which should be

compared

to the bound

lxx)

m 0.11895 derived

by

Nishimori and Ozeki

[10]

and to the

numerically

obtained value of

lxx)

m 0. I1167.

Inserting

_our bound into the

original proof already

extends its

validity

from A

m 1.67 to A _m 1.61.

Adding

the

significantly improved

variational

energies

we calculated in section

4,

the

original proof

becomes valid for all

anisotropies

A _m 1.52 and 0 _w A _w 0.19.

Numerically evaluating

the

integrals r~(a)

and

inserting

the lower bound

lxx)

~ m 0.13942 _we derived in section

3,

the variant of the DLS method which _we derived

in section 2 then leads to

proofs

of LRO in the

x-component

for all

anisotropies

0 _w A _«

A~~~(a )

and in the

z-component

for all

anisotropies

Am

A~~(a

with the

limiting

24

z6

11

₂₈

30

32

0 Z 4 6 8 1-O

1i

Fig.

2. Variational

energies

for the XY-like

anisotropies.

The centre lines show the

energies

obtained for the test state 4~~

) by dividing

the infinite lattice into

plaquettes

of different sizes. The uppermost line

corresponds

to the N6el-state

(in x-direction)

and the lowermost line

corresponds

to exact lower bounds calculated

using equation (23).

The interaction

strengths

_were chosen

according

to

(25)

and _a 5 _x 4

plaquette

was used.

z5

30

35

11

₄₀

45

50

55

1-O I-Z 1-4 1-B 1.8 Z-O

A

Fig.

3. Variational

energies

for the

Ising-like anisotropies.

The centre lines show the

energies

obtained for the test state

4~~) by dividing

^{the infinite lattice into}

plaquettes

of different sizes. The uppermost ^line

corresponds

to the N6el-state (in

z-direction)

and the lowermost line

corresponds

to exact lower bounds calculated

using equation (23).

The interaction

strengths

_were chosen

according

to

(25)

and _a 5 _x 4

plaquette

was used.

values listed in table I.

Finally optimizing

the value of _{a, we} find the widest _range of

validity

for small

positive values,

_{as was} to be

expected

since

(6)

has

previously

led to better limits

than

(5).

We have thus achieved _our

goal

^of

extending

the

proof

of LRO from Am1.67 to

A _m 1.47 and from 0 WA _< 0.13 to 0 WA w

0.22, closing

30 9b of the

remaining

_{gap on} the

Ising-like

side and

extending

the scope of the

proof by

70 9b _on the KY-like side. Further

significant improvements using

this method _seem

quite unlikely,

and

although

_we have been able to

slightly improve

the method

itself,

results for the

isotropic Heisenberg

model _can still not be

obtained,

_even if _we insert the numerical estimates for the

nearest-neighbour

correlation functions. As

Kennedy

_et al.

[4]

^{and Nishimori and Ozeki}

[10]

^have

previously pointed

out, a new method is

required

to treat the

isotropic

case.

Acknowledgements.

We wish to thank the Rechenzentrum der Rheinisch-Westiilischen Technischen Hochschule Aachen for its kind

hospitality during

the numerical calculations. This research _was

performed

within the program of the

Sonderforschungsbereich

341

supported by

the Deutsche

Forschungsgemeinschaft.

Appendix.

Dividing

the square lattice into 5 _x 5

plaquettes,

_we obtain the

following

values

Io

and

T,

which _are

required

for the calculation of the _energy of the variational _{test state}

~P~ in section 4 _:

Io(A)

=

= 383504 A^~~ ₊ 1931008 A^~~ ₊ 4120288 A ^~° ₊ 5074336 A ^~~ + 4085352 A ^~~ ₊ 2284144 A ^~~

+ 911872 A ^~~ ₊ 261872 ^~° ₊ 53645 A^~₊ 7640 A^~ + 718 A^~ + 40 A ^~ + l

(42)

7~(A)

=

=

(-

1164068293120 A^~~ 9225631229952 A ^" 32850592375808 ^" 69738691879936 A^~~

98317887695360 A^~° 96181079215104 A^~~ 65011188580096 A^3~ 27611636833792 A ^3~

3621561153088 A ^~~ ₊ 4137375844608 A ^~° ₊ 3339275955712 A ^~~ ₊ 1086767643200 A^~~

49682207392 A^~~ 247925155392 A^~~ 148968844112 A ^~° 56118842240 A^~~

15451961466 A ^'~ 3256218376 A '~- 532373904 A ^'~ 67330632 A ^'°

6480956 ^~ 459768 A^~ 22688 ^~ 696 A ^~

10)/4

(43) T--(A )

₌

= 13156864 A ^~~ 50016000 A^~~ 79563392 A ^~° 72818624 A ^~~ 44628544 A^~~

20555744 A ^~~ 7838816 A ^~~ 2554560 A ^~° 671096 A^~

129016 A^~ 16448 ^~ 1224 ^~

40)/4 (44)

T~_ (A

=

= 4602048 A^~~ ₊ 21241088 A ^~' ₊ 41202880 A ^'~ ₊ 45669024 A ^'? ₊ 32682816 A ^'~

+ 15989008 A^~~ ₊ 5471232 A ₊ 1309360 A^~ + 214580 A ^? ₊ 22920 A^~ ₊ 1436 A ₊ 40 A

(45)

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