New variational series expansions for lattice models

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New variational series expansions for lattice models

M. Kolesík, L. Šamaj

To cite this version:

M. Kolesík, L. Šamaj. New variational series expansions for lattice models. Journal de Physique I,

EDP Sciences, 1993, 3 (1), pp.93-106. �10.1051/jp1:1993119�. �jpa-00246715�

J.

Phys.

I France 3

(1993)

93-106 _JANUARY 1993, PAGE 93

Classification

Physics

Abstracts

05.50 75.1011 64.60

New variational series expansions for lattice models

M. Kolesik and L.

(aiuaj

Institute of

Physics,

Slovak

Academy

of

Sciences,

D6bravsk£ cesta 9, 842 28

Bratislava,

Czechoslovakia

(Received

22 June 1992,

accepted

in final form 27

August 1992)

Abstract For the symmetric two-state vertex model _on the honeycomb lattice _we construct a series

expansion

of the free energy which, at finite

order, depends

_on free _gauge parameters.

We _treat these gauge parameters _as variational _ones, and derive _a canonical series of classical

approximations

which _possesses the property of coherent

anomaly.

As _a test model _{we use}

the vertex forinlllation of the

Ising antiferromagnet

in _a field and, within the

coherent-anomaly

method, determine with _a

high

_accuracy its critical frontier and exponent _7. Numerical checks

on the constancy ^{of critical} exponents

along

the

phase boundary

_are

presented,

too.

1 Tut i,oductiou.

Rigorous approaches

to lattice systeiiis are

usually

based _on series

expansions

in powers of

some siuall

paraiiieters [1-3].

^{Such series} are, in

general, only slowly convergent

^and _so ^much effort has been done _to accelerate the convergence, e-g-

by introducing

_a variational

parameter

into series

expansion [4].

Their

extrapolative analysis requires

the

expected

form of the critical behaviour. On the other

hand,

consistent closed-foriu

approximations

based _on

physical

as-

sumptions

and

self-consistency requirements,

such _as the

mean-field,

Bethe

approximations

and their extensions

including

fluctuations inside finite

clusters,

while

providing qualitatively good phase

_structure,

give only

classical

singularities.

This

deficiency

is removed in the remarkable

coherent-anoiualy

method

(CART)

_[5]

inspired by

the finite-size

scaling theory _[6]. Here,

the nonclassical exponent ^is estimated from _a coherent

anomaly

which _appears in the coefficients of classical

singularities

obtained

by systeiiiatically increasing

the

approximation degree.

For

the 2D

Ising ferroniagiiet,

the

problem

of

a suitable choice of canonical

approximation

series has been

intensively investigated

in references

[7-1Ii.

The best results have been attained

by using

the multi-effective-field

theory [li]

under _solve restrictions _on the combination of effective

fields.

The aim of this _paper is to _cons truct _a canonical Series of

approxiniations

which _possesses the

property

of coherent

anoiiialy

and _can be

applied

to _iuore

general systems having

_a nontrivial

phase

structure. As _a test niodel _{we use} the

Ising antiferroiuagnet

in _an external

field,

formu-

lated _on the

hoiieycoiub lattice,

which ha~s been _a

long

tinie the _source of _many contradictions

JOURN~L DE PHYSIOUE i T 3, N' I, JANUARY 1993 4

94 JOURNAL DE

PHYSIQUE

I N°1

[2]. Recently,

it has been shown that its critical fi.ontier _can be

expressed

with

a

high

_accuracy

as a linear combination of fuiidaiueiital invariaiits of the _gauge transformation constructed from statistical

weights

^of a related vertex model

[13].

^{Our method} is based _{on a} close relation

between variational series

expansions

and closed-form

approximations. Using

the

technique

of

a

generalized weak-gray)h

transformation

[14-16],

we formulate the

spin system

as a

symmetric

vertex model with free _gauge

paraiiieters.

^Iii the series

expansion

of the free energy of the _ver-

tex

model,

_we treat the gauge

paraiiieters

_as variational _ones and _recover in the lowest order of the

expansion

the Bethe

approxiiiiatioii (Sect. 2).

~fith

increasing systematically

the order of variational series

expansion

the internal

consistency

of the

theory

remains

maintained,

the calculated critical

points

_converge to the _{true one} and the series has the

property

of coherent-

anomaly scaling.

The consideration of _a "reference"

subspace

of model

parameters

with the known critical

point

allows _us to

deteriiiine,

within the CAM

theory,

with _a

high

_accuracy the critical characteristics and to

present

iiuiiierical checks _on the

constancy

of critical

exponents along

the

phase boundary (Sect. 3).

The iiiethod _can be

directly applied

to

symmetric

vertex

models of which

special

_{cases are}

briefly

discussed in section 4.

2. Variational iuetliod iii the lowest order of series

expansion.

We _are concerned n.ith the

systeiii

of

Ising spins

_{si =} +I which _are located _at sites I ₌

1,2,. ,N(-

_cro) of t-he

honeycoiiib

lattice n,ith

cyclic boundary

conditions. Its

partition

function reads

Z(fl,

h

=

~j

exp

(fl ~j

si sj ^{+ h}

~j

si

),

I

lsl (I,iJ I

where the

nearest-neighhour coupling (inverse temperature) fl

and the

applied

field h _are dimensionless.

In order _{to iiiap} the

[sing

niotlel onto the vertex iiiodel _we decorate each

edge

of the

honey-

comb lattice

by

_{a new i,crtex.} Let the

resulting edge fragments

_can be in _one of two distinct

states _s E

(+, -).

~vith each three-coordinated vertex of the

original

^lattice we associate the

vertex

weight w(si,

_s2,

s3)

which

depends

_on the state

configuration

of incident

edges according

to

w(si,

_s?,

s3)

_"

exp(h

for _{si " s2 " s3 "} +,

=

exp(-h)

for _si

= s2 = s3 " -,

(2)

= 0 otherwise.

Here,

the admissible

configurations

with all tliree incident

edges

in the _same

+/-

state are

identified with the

up/<lown

state of the

spin

located at the

corresponding

vertex. In order to reflect the

coupling

^of

nearest-neighbour spins,

the elements

P(s, s')

of the

weight

matrix for

decorating

two-coordinated vertices _are cliosen _as follows

eP e-P

i~ _"

~-# eP (3)

Hereafter,

the columns

(rows)

of I _x 2 iuatrices _are indexed from left _to

right (from

_up to

down)

as +, ^The

partition

function of the

resulting system,

defined

by

Z =

~j fl(n,eights) ⁽⁴⁾

lsl

N°I NE~V VARIATIONAL SERIES EXPANSIONS FOR LATTICE MODELS 95

with the summation

going

_over all

possible edge configurations

and the

product being

_over all

vertex

weights

in the

lattice,

is

evidently

identical to that

given

in formula

(I)

for the

Ising

model.

The P-matrix _can be n,ritten _as

a

product

of two matrices

P "

VI _V2, (5)

where the

subscripts

indicate that the

underlj,ing honeycomb

lattice _can be

composed

of _two

nonequivalent

sublattices I and 2

(this

is the _case of the

low-temperature phase

of the anti-

ferromagnet).

In

general formulae,

which remain

unchanged

after the

exchange

of

subscripts

I and

2,

_p will denote I _or 2 and ~p,_VI =

[1,

2] or

[2,1],

_e-g-,

Vp

₌

Vi

_or

V2, Vpvv

₌

ViV2,

etc. We propose the

following paranieter12ation

of the V-iiiatrices:

(I) ferromagnet (fl

>

0)

~(~2p ~-2fl)1/2

_(~2fl

~-2fl)1/2

Vi

_"

V2

₌

j

,

(fia)

~

e~ e~~z e~~ e~z

v =

e~(I

₊

x~) 2e~~z; _(fib)

(it) antiferromagnet (fl

<

0)

~

(~-2fl ~2fl)1/2 (~-2p ~2p)1/2

VI

_"

fi

,

(7a)

~

e~ e~~z2 e~~ e~z2

~

j~-2fl ~2fl)1/2 j~-2fl ~2fl)1/2

~

i ²

~ ~~

~~~~

_(eP e-P~~) _(e-P ePz~)

v =

e~~(xi

₊

_x2) e~(1

+

ziz2). (7c)

There exist also other

siiiipler paranieterizations

of the V-matrices but this _one

provides

the most

compact

formulation of the lowest order of the variational series

expansion presented

in this paper.

Grouping

the

triplets

of V-matrices attached _to the

corresponding

^{nodes into} new

weights,

_we omit the

decorating

nodes and obtain the vertex

system

^defined on the

original

lattice with the follo&&>ing ^vertex

weights

tiff(Sl>S?,53)" ~j ~l'(Sl,S~)~tf(52>S~)~l'(~3,S~)~(~~'~~,S~l' (~)

,i,>,>

With

regard

to the

explicit

foriu of

ui(si,

_b2,

s3) (2 ),

^these statistical

weights

_are

independent

of the

periuutation

of _{si, s2, s3.} Let _{us use} the notation _a, b, _c, d for the

symmetric

vertex

weights corresponding

to the vertex

configurations with, respectively, three, two,

_one, zero incident

edges

in the state

(+). They

read

a~ =

l§,(+,+)~e~'+ _Vp(+, -)~e~~', (9a)

b,,

=

V»(+,+)~~l,(-, ^+)e~'+ _Vt~(+, -)~V»(-, -)e~~', (9b)

C» =

V»(+, +)V»I-, +)~e~'+ V»(+, -)V»(-, -)~e~~', (9C)

dj,

=

(,(-, +)~e'~

+

V~(-, -)~e~~' (9d)

96 JOURNAL DE

PHYSIQUE

I N°1

The free

parameters

_zp ^{included in} elements of V-matrices

play

the role of real-valued

(gauge)

parameters

^{of the}

generalized weak-graph

transformation

(introduced by Wegner [14]

^and _ap-

plied

in the

general12ed

form to the

present

^model

by

Wu

[IS, 16])

^{which leaves the}

partition

function of the

syiunietric

vertex iuodel invariant.

Now _we

explain

how to use the _vertex foriiiulation of the

Ising

^{model for}

studying

its

thermodynamic properties.

Let _{us suppose} that the

syniiuetric

vertex

weights (9a-d)

fulfil the

inequalities

_{a~ >}

([b~[, [cp[, [d~[).

The

ground

state of the system ^is then

represented by

the

configuration

with all

edges

in the

(+)

state. Its contribution to the statistical _sum is

(al a21'~/~

and the

corresponding (diiuensionless)

free energy per site reads

lo

_"

log(aia2). (10)

Although

the exact free energy cannot

depend

_{on gauge}

parameters,

its finite-order

expansion (which

is the _case of

(10))

I.s a function of z~. ^{life treat} z~ as variational

parameters satisfying

the

stationarity

conditions

0xp(~fo)

_" 0

(II)

and

easily

find

using (6), (7), (9)

that these conditions _can be identified with

b~ = 0.

(12)

For the Bethe lattice which has _no

cycles,

the

change

of _a finite number of

(+) edge signs

in the

ground

state

gives

_a

vanishing

contril)ution to the statistical _sum under the condition

(12).

Thus,

tile

systeii~

is fro2eii in tile

ground

state and its free _energy is

exactly given by (10).

^The

relationship

to the Bethe

approximation

is better _seen fi.oiu the

explicit

form of

equations

for

z~'s implied by

conditions

(12).

For the

ferroinagnet

and the

antiferromagnetic phase

^with

equivalent

sublattices

(xi

_{= z2 =}

z)

_we ol)tain

z~e~~+~' z~e~+~'+ xe~~~' e~~~~'

= 0.

(13)

For the

antiferroniagnetic phase

with

nonequivalent

sublattices I and 2 gauge

parameters

xi

#

_{z? are} the (lifferent _roots of the

secon(I-degree polynoiuial x~(e"+~~'

₊

l)

+

x(e" e~")

₊

_(e~~~~~'

₊

₁₎

= 0.

(14)

Having

all real-valued

pairs

of sublat,tice roots

(z(J, x(~),

the free energy is determined

by

lo

" nijx

(- _{fo(x(~~, z(~)j} (15)

1

We _now discuss

briefly

the structure of roots because its relation to the

phase

structure is similar in

higher approximations.

The fi.ee energy of the

ferromagnet

is

singular

_on the

line h

=

0,fl

>

fl~.

On this line

(except

for

fl

₌

fl~),

there exist two "dominant" roots to

equation (13) ^z(~) # xii ),

such that

fo(z(°)

=

fo(xii)),

and _so the first-order derivative of

lo

with

respect

to h is discontinuous. At the critical

point

_fl~

=

arctanh(1/2), ^z(~)

is

equal

to

z(I)

and _a second order

phase

transition takes

place.

For

p

<

fl~,

there exists

only

_one real-valued solution _to

equation (13).

As to the

antiferroiiiagnet,

its

phases

_are determined

by

the

sign

of the discriminant D of the

second-degree polynomial (14):

D =

e~~~ 3e~~ 4e"(e~~'

+

e~~~')

^6.

(16)

N°I NEW VARIATIONAL SERIES EXPANSIONS FOR LATTICE MODELS 97

2.0

I I

I

o-o

o-o i

h /lfl

Fig.

I. The phase

diagram

of the

antiferromagnetic Ising

model. The dashed and solid lines

cor-

respond

to L

= 0 5

(Bethe-approximation)

and the

extrapolation

of L

= 5 -19 data,

respectively.

Points

(+)

obtained

by

finite-size

scaling analysis (Ref. [13])

_are

given

for comparison. The _zero field critical points coincide with those of the

Ising ferroinagnet.

In the

high-teiuperature region

with

equivalent

sublattices

(D

<

0),

the roots to

equation (14)

are not real and _one real root to

equation (13)

determines the free energy

through

relation

(15).

In the

low-temperature phase (D

>

0),

^the doiuinant role is

played by

real roots to

(14)

xi

#

_z2

Owing

to the

decomposition

of the P-matrix into

unequal

matrices

Vi

and

V2 (5)

the sublattices I and 2 _are

nonequivalent:

the order

parameter

^is

represented by

the difference of sublattice

magnet12ations

_{nil )ii?.} ^At the

phase boundary D(li~, h)

= 0

(see Fig. I),

the roots

xi " z2 to

equation (14)

coincide with _one root t-o

equation (13)

and the

system undergoes

a

second-order

phase

transition.

3. Variational inetliod in

liighei,

orders of series

expansion.

In this

section,

_we construct _a

systeiuatic

series of

iiuproved approximations by taking

into

account

higher

ternis of the

expansion (10).

The consideration of the cumulant contribution

of successive

excitations, generate(I

fi.oiu the _sea of

(+) edges by changing

the

(+) edge signs

on a connected lattice

graph,

^enables one to obtain the series

expaiision

of the free energy in powers of the siuall

quantities _(bj, lap ), _(cj, la~ _{), (d~} lap ):

IL

₌

log(aia~)

+

f _II')

(~", ^{~", ~" (17)}

a~ a~ a~

Here, f(')

_is

a function of the lth order in

arguiuents.

The

explicit

forms of

f(')

_up to1

= 19 _can

be

straightforwardly

deduced fi.oni foriuulae

(A2), (A4-7) presented

in

Appendix.

In

particular,

98 JOURNAL DE

PIIYSIQUE

I N°1

we have

/(1)

~ ~~

/(2)

~

3

(bl) _(b2)~ ^/(3)

~

3

(b2)~(~l

~

(bl)~(~2)j

~ _{~ll a2} 2 _{a2 al al a2}

/(4)

~ ~

bi

b2 _{Cl C2} IS

bi

^~ b2 ^~

~

l

(

_b2

~

(di

^~

^bi

~

d2

~ll °2 al ~l2 4 _{al °2} 2 _{a2 al al a2}

and _{so on.} The first nontrivial

contribution,

which has _no

analogy

for the _case of the Bethe

lattice,

is the

hexagon

contribution

(cilai)~(c21a?)~/2

in

f(~l.

_Number L is the _upper limit of the number of vertices _{on a} connected lattice

graph

with at least

one incident

edge

in the

(-)

state and iii fact

represents

^{tile cluster s12e considered in} a

given approximation.

We have conceiitrate(I _on the

antiferroiuagnet

in _a field and used the

change

of gauge vari- ables

d ₌

)(xi

⁺

^z~),

a ₌

jlzi z~), (18)

in order to reflect the

syiumetry fL(d, a)

₌

fL(d, -a).

We have

investigated numerically

the

stationarity

conditions

0~(- IL

_"

da (- IL

_" 0 for L ₌ 0 19. The structure of real solutions is the _{same as} for the Bethe _case L

= 0. For _a

given h,

there exists either _a

single

solution located _on the d-axis

(a

=

0)

_{for [fl[} < [/J~[ or three solutions distributed

symmetrically

with

respect

to _{a =} 0 for [fl[ > [fl~[. In the

low-teiuperature phase

two

equivalent

dominant solutions

are those with _iion2ero

«-part.

At tile critical

teiJJperature

all t)Jree solutions coincide

and,

_as

a consequence, the second-order

phase boun(lary fl[(h)

is determined

by

the condition

°aa(-fL)lp=pj(hj

_" ^0.

(19)

In order _to calculate

theriiiodyiianiic quantities,

_we introduce in formulae

(9a-d)

_an

alternating

field

hp acting

_on sites of the

corresponding

sublattice of type _p ^and at the end of the calculation

we put

hi

_"

h2

₌ h. It is _easy to show that the order

paraiiieter

and the

(dimensionless)

staggered susceptiiiility

_j

=

(fii~

_hi +

0h~h~ 20h,h~ )(-IL

exhibit the classical

singularity

at the critical

point

and _can be

expressed

_as

nit »i- - >,iL

~flii'i~ fli~l'

~~~~/~~~~/~ j««~/~~~~/~ 1/2 ll])

~j~~ = (@~h~

IL ~ah~/L)

~

~( f~)2 @aaaa/L~bb/L

X "

iL

~ )c _jj~~' ^(21~)

L

'~

0~~i)i~j~~~ ^{ia~))~~~~L ~~~~~}

Here,

all derivatives _are taken at the critical

point

and at

hi

₌

h?

₌ h. As indicated at the

beginning

of this

section,

for L ₌ 0 5 the obtained numerical values of

fl[(h),mL, _gL

are identical _to those

given by

the Bethe

approximation.

For

higher

values of

L,

the critical

couplings fl[(h)

_converge to the _{true one}

fl~(li).

This

suggests

that _our successive series of

approximations

is canonical and has the property ^of

coherent-anonialy scaling.

Before

proceeding fiirther,

_we ret,iew

briefly

tile CART

theory

_[5]

adapted

to _our

problem.

At the critical

point flj (h

^{which is}

yielde(I by

the

approximation considering

cluster sizes <

L,

N°I NEIV VARIATIONAL SERIES EXPANSIONS FOR LATTICE MODELS 99

Table I. Calculated

critiial points

_fl~ arid the

coherent-anomaly _exponent

_t~ ^~~~ critical field calculated in reference

[13] by

finite-size

scaling analysis

for the

Ising coupling given

in the second column. H~e took ~~~h _as

input

field values and calculated

fl~

^and t~ shown in the rest of table. ^~~~ values of fl~ ^and t~ obtained from

large-L _asymptote (22)

and

(23b).

^~~~~~~~

values of

li~

^obtained from L ₌

G,. ,19

data inserted into

equation (25) (see

the

text):

^~~~

the best estimation of

fl~,

^~~~ the worst estimation of

fl~,

^~~~ the _mean value of

li~.

~~~h fl~ ~~~fl~ ^~~~fi ~~~fl~ ~~~fl~ ~~~fl~

0.0 -0.fi5847 -0.65448 0.729

0.582431408 -0.7 -0.G9G38 0.73i -0.700045 -0.70034 -0.70019

1.119884213 -0.8 -0.19103 0.737 -0.800110 -0.80101 -0.80056

1.520604370 -0.9 -0.89745 0.740 -0.900142 -0.90148 -0.90081

1.875990455 -1.0 -0.99772 0.742 -1.000156 -1.00182 -1.00098

2.530154218 -1.2 -1.19800 0.744 -1.200164 -1.20220 -1.20117

3.458129780 -1.5 -lA9817 0.745 -1.500175 -1.50242 -1.50128

the correlation

length

of the true systeiii ^is

proportional

to L.

Consequently,

_near the _exact critical

point fl[(h

has the

asyiuptotic

foriu

fl[ (h

-~

fl~(h)

+

a(h)L~

^~/"

(22)

The coefficient

a(h (lepends

_on the

particular

choice of

approximation

series and _on the critical

exponents

for _a

given

value of I). The mean-field critical coefficients iL and

gL diverge

_as the cluster size becoiues

large according

to

>I> L ^-~

L~/", _jL

-~

L~/" (23a, b)

By coiubining equations (22)

and

(13a, b),

_we arrive at

"'~ "

[Pill>) flat'>)l~

^{' ~~~~~}

'~ ~

~j[(11) fl~(lJ)]~ ^~~~~~

The

exponents

-# arid ~,

represent

the corrections to the classical

exponents 1/2

and

I,

_respec-

tively.

Predictions of the CART _are not, in

general,

in

perfect agreement

^{with obtained data} for mL. That is

why

_we restrict ourselves to the

study

of coherent

anomaly

for the

staggered

susceptibility.

We first discuss the results estiiuated fi.oiu

large-L asyiuptote

of

fl[(h) (22)

and

iL (23b) (see

Tab.

I) by using

linear fits

(with

_{v =}

I)

in

appropriate

scales

(the plots

of

fl[(h)

_versus

I

IL

for h

= 0 and I-119884213 _are

presented

in

Fig. 2).

In the considered

large

^{field interval}

0 < h <

3.45812978,

all the estiiuations of critical

couplings fl~(h)

lie within _{an error} of 0.6

percent

of the exact and most reliable values of critical

couplings [13]

ⁱⁿ

spite

of _a

relatively large dispersion

of data

points

the correlation coefficient in the

least-squares fitting

_r

-~ 0.74 0.82

([r[

₌ I when

points

lie

exactly

_{on a}

line).

The

exponent

_1~ ^exhibits a

slight dependence

on h and

changes

from 0.729 to 0.745 _as h _goes from 0 to 3.45812978.

100 JOURNAL DE

PHYSIQUE

I N°1

-0.45 145

D D ~

-0.50 damn°°°° ° °

-o.55

~~~~

140

-o 60 h"°.°

,"~"~'~~~"' ^i~~i~

/~~

^~

_p«~

L

135 -o.65

~°.~° ,a

h= I. 19884213

,,"'

_~°

~,~-~--~K

-o.75

~

~AA«~

-0.80 125

0.00 0.05 0. 0.

IL

Fig.

2.

(b)

The critical

points fl[

for two values of external field h.

(Q)

The

corresponding

difference

6fl[

=

fl[(h

=

1.11984?13) fl[(0).

In

general,

the numerical data fit better the formula

(24b)

than the

large-L asymptote

(23b), however,

the value of the

exponent

_1~ ⁱⁿ

(24b)

is _very sensitive to the assumed result for

p~(h).

That is

why

_,ve

iiuprove

the _accuracy of the determination of

fl~(h) by considering

the

"reference h

= 0

system"

with the known value of the critical

coupling fl~(0) =-(arcsinhvi)/2.

The subtraction of

equation (22)

with h

= 0 and the _one with _{a nonzero} value of h

gives

Afl[(h)

₌

fl[(o) fly(h)

+~

fl~(0) fl~('4

+

la(0) a(h)lL~~/" (25)

As shown for h ₌ 1.119884213 in

figure 2, Afl[(lt) is, except

for the Bethe _case L ₌

5, practically independent

of L with _a

dispersion

lower than I

percent

of its _mean value

(we

note that since

Afl[(h) represents only

_a small part ^of

fl[(h),

the

dispersion

of

fl~(h)

itself is _even

substantially siualler). Consequently,

the

plots fl[(0), fl[(h)

_are shifted

by

_a constant

value

fl~(0) fl~(h)

and

a(0)

-~

a(h).

Since the a's

correspond

to the _same canonical series of

approximations

this also _means that the critical

exponents

coincide with _a

high

_{accuracy on}

the whole

phase boundary.

In the last coluiuns of table

I,

_we

present

for various values of h the

minimal,

ma~~imal an(I _mean values of

fl~(h

which _are deduced from 14 data

(flj (0), fl[(h)) _(~

~

inserted into

equation (25)

with

neglecting

the

L-dependence

of the

right-hand

side. The best

(worst)

estiniations of

fl~(h)

are consistent with the most reliable values

[13]

to within 0.02

(0.16) percent

with the evident

tendency

to the

iniproveiuent

_as L increases and

fl[(h)

approaches

to the _true critical

coupling fl~(h).

The best estimations of

fl~(h)

_are found in the

highest 19th

order and will be used in ii,hat follows. ive

eiuphasize

that the above

analysis,

which eliiuinates the influence of _a

relatively, large dispersion

of the

input data,

_can be

applied

to the _case of the critical frontier with

varying

critical

exponents,

too.

ilaving

the

precise

location of

fl~(li ),

_we can use

equation (24b)

to estimate

exponent

t~. ^For h = o the

logarithius oft L's

_are

plotted against log~fl[(0) fl~(0)]

in

figure

3. The

points

do

N°I NEW VARIATIONAL SERIES EXPANSIONS FOR LATTICE MODELS lot

not lie

exactly

_{on a} line because the data for lower values of L _are located in the

region

where further terms of the Laurant

expansion (24b) play

the role. To eliminate the contribution of these terms, we have

successively

omitted the data with L _< N

(N

=

5,6,. ,17)

and calculated the

exponent

t~ from data with L

=

N,

N

+1,...,19.

The

tendency

is shown in

figure

4. With

increasing

N

(I.e. taking

into account data _nearer the critical

point),

the value of t~

decreases,

_except for small local

fluctuations,

towards the _exact

one 0.75. The correlation coefficient in the

least-squares fitting

_r

changes

from -0.9998

(N

=

5)

to -0.99996

(N

₌

17).

A

slight dispersion

of the last three values _{of t~ E<}

0.749,0.755

_> is due to a

relatively

small number of considered data. In

spite

of

this,

the _accuracy of the determination of critical

exponent

₇ ^is

comparable

to that attained in reference

[12]

^{for the}

Ising ferromagnet using

_some

special

choices of canonical series of

approxiiuations.

The

plots

of _{t~ versus} N _are

practically

independent

of the

applied

field h _as shown for h

= 1.119884213 in

figure

4. This fact confirms

again

^the constancy ^{of the critical}

exponents along

the

phase boundary.

.2

o

o

o°o

n

0.8

/5 ~

ill

o

~/

~J~

O

~

0.4 °

$

o-o

-4.0

jog _(fl~ ~fl~

Fig.

3. The

logarithms

of the

coherent-anomaly

coefficients iL

Plotted

_versus

log(fl) fl~)

for h = o.

4.

Concluding

remarks.

We have

proposed

_a method for

investigating

^lattice models which consists in _a combination of the series

expansion

with the variational

approach

and

applied

it to the

Ising antiferromagnet

in _a field formulated _on the

honeycoiub

lattice. The method

provides

_a canonical series of

approximations

which has the

property

^{of coherent}

anomaly

and allows _one to determine with

a

high

_accuracy ^the

phase diagraiu

and critical

exponent

_7. Two _more features _are

worthy

to be noted:

(I) using

the gauge transforniat.ion

technique presented

in

Appendix

_{one can} derive the series

expansion

_up to

high

orders without

necessity

of

enumerating

_a

plethora

of lattice

lo2 JOURNAL DE

PHYSIQUE

I N°1

0.80

o.79 '

K

0.78

~'

fi'

all

^0.77

~9i,

~

@JJ~~-4

o.76 '

n

o.75

0.74

Fig.

4. Analysis of tlte coherent-anomaly exponent _~b for the _zero field

(o)

^{and for h} ₌ ^1.11984213

(b).

_~b is

given

_{as a} function of N the minimal order of

approximations

included in the linear

regression

data for

calculating

_~b.

graphs

which _possess vertices connecte(I with _a

single neighbour; (2)

the structure and the number of real solutions to

stationarity

conditions do not

change

^with

increasing

the

degree

of the

approximation.

The

study

of the

syiunietric

vertex model _on the

honeyconib

lattice indicates that

(2)

need not be

alwaj,s

the _case. ~rhis

systeiii

^is

equivalent

to the

Ising

^iuodel with either

ferromagnetic couplings

^and real-valucd field

(f-reginie)

_or

antiferromagnetic couplings

and _pure

imaginary

field

(a-regime) [16, 16].

^The

a-regiiue

deserves

special

attention because it is not known whether the vertex iuodel in such _a

regiiue undergoes

_a

phase

transition. The

application

of the variational iuethod is _{now even more}

straightforward.

^{Instead of}

parameterizing

the

Ising

interaction iuatrix P _n-e insert

directly

the gauge transforiuation of vertex

weights

of

type (A2)

into the series

expansion (17).

The nuiuerical results indicate that the structure of real solutions to the

st.ationarity

condition is

quite

different in the

a-regime

in

comparison

with the

complementary, f-regime

^of the vertex nio(lel. There _are

generally

_more

solutions,

but the

dominant _one

(wliicli

turns out to be

syiunietric

_y~

=

y)

can be

easily

identified. It neither coincides with other roots _nor exhibits _a

(liscontinuity.

This

gives

the evidence for the absence of

phase

transitions iii tile

a-regiiue.

In order to demonstrate the

interplay

_aiiiong the solutions of the

stationarity conditions,

we have also studied t-lie iuononier-diiuer system formulated _on the

honeycomb

lattice. The

equivalent syiunietric

,,ertex model _possesses

only

two nonzero statistical

weights corresponding

to vertex

configurations

with _2ero and _one incident

edge

in tile

(-)

state: _{a =}

exp(-li),

b

= I.

In the limit of _zero

teiuperature fl

- cro, it reduces to the

close~packed

dimer model

[17].

^The

mapping

onto the

Isiiig

ino(lel leads to _a rather

special liniiting antiferromagnetic

_case which is difficult to treat

(lirectly jib].

In the Bethe

(zerc-order) approximation,

there exists

only

_one root to the

stationarity

condition _{y =}

II /.

_{The number of roots increases} with

increasing

the

approximation

order L.

N°I NEW VARIATIONAL SERIES EXPANSIONS FOR LATTICE MODELS lo3

Fortunately~

_{one can}

identify

the dominant _root

unambiguously.

All the roots _are distributed in _a not too wide interval

(depending

on the

temperature)

and the free _energy taken _{as a}

function of the variational

parameter displays

_a

plateau

in the middle of this interval. It turns out that outside of this

plateau

the series fails to converge ^at low

temperatures

and _so the

corresponding

roots _are to be omitted.

Applying

the usual minimum criterion for the free energy one finds the

physical

solution _among the rest of roots. The solution chosen in such _a way

slowly glides

froiu the zerc-order value _y

=

I/vfi

_{at infinite}

temperature

to

higher (lower)

values for

fl

_{- cro}

(fl

-

-cro)

without coincidence with other roots. This is in

agreement

^with the fact that there is _no

phase

transition in the considered monomer-dimer system

[18].

Our results

are illustrated in

figure

5 in which the free

energies

_per site

IL

_are

plotted against fl

for the l3ethe

approximation

L ₌ 0 -,§

(dashed line)

and the

highest-order

_approx-

imation L ₌ 19

(solid line).

These

approximations

differ

significantly

from _one another

only

in the

low-teiuperature region.

life have calculated the concentration _n of the b-vertices _as

a function of

fl,

too. The

plot

is shown in

figure

5.

only

for L

= 19 because there is _no visible difference between t,he

approxiniations

_on this scale. The check of accuracy is _prc- vided

by

the

2ero~teinperature

value of the fi.ee energy

(entropy)

which is

exactly

known to be

f(cro)

₌ 0.lG15.

([or

foriuulae for the exact solutions of the

close-packed

dimer models

see

[19]).

Our estimation

-f19(oJ)

= 0.1591 lies within _{an error} of I-S

percent

^of the exact value. This is _a consiiierable

improvement

of the value

given by

the Bethe

approximation

fo(cro)

₌ 0.1438.

.50

n i .oo

o.50

~~9

o.oo

io.oo

~

Fig.

5. The monomer-dimer model _on the

honeycomb

lattice. The dimensionless free energy

f

and the concentration

n of dimer vertices in the-order L =19 _are

plotted by

^solid lines. The broken line represents ^the free energy in the zero-order

(Bethe) approximation.

In

conclusion, owing

to the relative

siiuPlicity, consistency

and _accuracy the

proposed

vari- ational series could

gi;e

_a

trustwortliy picture

of further vertex iuodels with unknown

phase

lo4 JOURNAL DE

PIIYSIQUE

I N°1

structure,

^such as the

symmetric

vertex model _on the _square lattice which _can be

mapped

onto the

Ising

model

only

_{on a} restricted manifold in the _space of the

symmetric

vertex

weights [20].

Appendix.

In order to derive the series

expansion

for the

symmetric

vertex model _on the

bipartite honeycomb

lattice with vertex

weights

_a~ >

[bp[, [cp[, [dp[,

we associate with each lattice

edge

the

identity

matrix written _as the

product

of two V-matrices

~ ₌

l VP

11

^{VP Al}

I + upup Vu I _Vu I

and group the

triplets

of V-iuatrices attached to the

corresponding

nodes into _new transformed vertex

weights

dt= =

(i

+

)y~~~/~

iat÷ + 3Yvbt÷ ⁺ 3YiCt÷ + Y?dt÷1

(A2a)

't=

=

ii

+

)yv)3/2

iYt=~t= + i~Yt=VP ~~~t= + _~Yt=Y~ ~Yv~~t= Y~~t=1

~~~~~

?t× ⁼

(i

+

)y~

~~/~

[via»

+

(YiYv

^2Yv)bt÷ +

(1

2Yt÷Yv)Ct÷ + Yvdt÷1

(A2C)

?t× =

(1

+

)y~~~/~

[via<, _3Yibt÷

+ _3Yt÷Ct÷ dt÷I

(A2=l)

Let _us consider that in the transfornied

I)icture

it holds

ip

= 0.

(A3)

This condition is consistent with the

following expansion

for gauge

parameters

a~

y~, ⁼

~ _{y['), iA4)}

1=1

where

yi~, being

of the lth order in

(bj, la~, ), (c~ la~), (d~ la~), satisfy

the recursion

y(')

₌ ^~~' _di,1 + 2 ~"

y$'~

+

~" ~j _vi'~

_yi~

~tl ~tl ~tl

j

(,+J=i-1)

2 ~»

~ _{yjj >up}

^~»

~ y[i)y[J)y[k) (As)

~»

j~i

"»

~.k~

i

(,+j=i-i) (,+j+k=i-1)

Having i~

₌

0, only graphs,vhich

_possess vertices connected with at least two nearest

neighbors

contribute to the cuniulant

expansion

of tile free energy

starting

from the

ground

state with all

edges

in the

(+)

state:

N°I NEW VARIATIONAL SERIES EXPANSIONS FOR LATTICE MODELS los

where the _non2ero

f(')

_which

are of the lth order in (Ep

lip ), (ip lip)

_are

yielded by

f(6)~~~(?t~)~(du)~

dt~ it~ ^'

(A7a)

(i°)=1.5(?t~)~(?v)~+15(?»)~(i~)4(l~j

ilt~

iv itp bv I (l~j £

'

(A7b)

" "

/~~~~

075(~")~(~")~

_dt~

_d~

^~

i$(?t~)~(?v)~(ip)(i~) _d~ ^{[ £ £}

'

(A7c)

~~~~ 3(~")~(~")~+o$(?1,)~(?v)~(i~)~i~j

^~

~ ~t~

iv

_it~

[ I I

'

(A7d)

t~ "

/(14>=6())~(?v)~+15(?t~)~(>~)6(l~jjl~j

~"

~" ii,

it~ it~

iv

~4s(?t~)~(du)~ ^ij,

(iuj~

~t~ it~

d,,

it~ ^'

(A7e)

/(15)

~ ^{5 5}

~( tl) (Cu)

~

ilt~

iv

^'

(A7f)

l~~~"~G(~)~(~")~+30(~")~(~")~(~t~)(~v)

~t~

~"

_~<'

iv

_it~

iv

195(~")~(~")~~~t~)~~~")~ ~~(?tl)~(Eu)~~ip)~(lp)~

il

i T

"

~

t

(A7g)

" ~" ~t~ _{" tl} au

~ a~

~~~~~

=

3.5(~"

)~(~"

~

+

4(~t~ ~ (iv

~ i~) (iv)

~ bt~ it

b~ I I I

" " " "

~3(?1,)~(iv)~ ^i~

~

vj~

~ _iv

it~~ itv

'

(A7h)

l~~~"3°.lG(~~)~(~~)~+IGG.5(~")~(~")~~~t~) ~")

b il it d

it~

^it

" " " " "

g(?t~)~(Fv)~(ij,j~(1)~

? ^~ i ⁶

j

^~

j

^~

~

£ £

^"

+14.5( ") (S)

^{t~ m}

~"

~" ~" ~"

_~t~

ilv

_ilt~

iv

~i~(?t~)~(iuj~ jd~j~(i~j~

jdp ~(i

~

~t~

~"

_bt~

[

^~

[)

©j

'

(~7iJ

~~~~~

=io2(?")~(?v)~+8~j5(?t~)~(i~j6(l~j(l~j

~ ~<, Au

dj, d~ it~ [

i~(?t~)~(i~)~~l~,j~~l~j~

₁ ⁴ ₁ ⁴

j

^~

j

³

~" ~"

_~t~

~" ~~~~il~~ ^k~~

it~~ it~~'

~~~~

106 JOURNAL DE

PIIYSIQUE

I N°1

~~~' '~~~~'

z =

(1)~(lv)~

⁺

(i)~(l»)~

a~ a~ au a»

Using (A5), (A4)

and

(A2),

_{we can express}

?~/d~

and

lj _lit~

as series in

b~la~, c~la~, _d~ la~

up to _an

arbitrary

order. Then

inserting

into

(AG), (A7)

_we obtain the series

expansion

for the

original

vertex iuodel.

References

[Ii Sykes hi-F-,

Gaunt D-S-, Roberts P-D- and Wyles J-A-, J. Phys. A 5

(1972)

640.

[2] Domb

C.,

in Phase Transitions and Critical Phenomena Vol. 3, C. Domb and M. S. Green Eds.

(Academic Press, London, 1974).

[3]

Onyszkiewicz

Z. and lvierzi)icki A., Phys. Lent. A i16

(198G)

^335.

[4] Bonnier B. and

Hontebeyrie M.,

J. Phj.s. I France1

(1991)

331.

[5] ^Suzuki

M.,

J.

Phj.s.

Soc.

Jpn

55

(1986)

4?05.

[6] Fisher M-E- and Barber

hi-N-,

Phys. Rev. Lent. 28

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1516.

[7] ^Suzuki

M.,

Ilatori AI. and IIU X., J. Ph_>'s. Soc.

Jpn

56

(1987)

3092.

[8]1(atori

M. and Suzuki _A>I., J.

Phj.s.

Soc.

Jpii

so

(1987)

3113- [9] ^flu X., Ilatori hf. and Suzuki if., J. Ph_i>s. ^Soc.

Jpn

5G

(1987)

3865.

[lo]

lI1l X. and Suzuki if., J. Ph_ys. Soc.

Jpn

57

(1988)

791.

[II]

hfonroe J-L-,

Phys.

Lett. A 131

(.1988)

427.

[12] ^Minami Il., Nonomllra

Y.,

Ilatori hf. and Suzuki M., Physica A 174

(1991)

479.

[13]^W1l

F-Y-,

Wu X-N- and B16te

II-W-J-,

Ph_ys. Rev. Lett. 62

(1989)

2773.

[14]

Wegner

F-J-,

Physica

68

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570.

[15] ^Wu F-Y-, J. A?ath.

Phys.

15

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687.

[16] ^Wu F-Y-, J.

Phys.

A 23

(1990)

375.

[17]

Ilasteleyn

P.~V., J. Alath.

Phys.

4

(19G3)

287.

[18] ^{lleilmann O-J- and Lieb}

E-H-,

Commmi. Jlath.

Phj,s.

25

(1972

190.

[19]

Nagle

J-F-, Yokoi C-S-O, and

Bhattacliarjee S-if-,

in Phase Transitions and Critical Phenomena Vol. 13, C. Domb and J. L. Lebowitz Eds.

(Academic Press,

London,

1989).

[20]

(amaj

L. and

Ilolesik

_M.,

Ph_ysica

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455.