Critical properties of the d-dimensional Ising model from a variational method

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Critical properties of the d-dimensional Ising model from a variational method

B. Bonnier, M. Hontebeyrie

To cite this version:

B. Bonnier, M. Hontebeyrie. Critical properties of the d-dimensional Ising model from a variational

method. Journal de Physique I, EDP Sciences, 1991, 1 (3), pp.331-338. �10.1051/jp1:1991135�. �jpa-

00246326�

Classification Physics ^Abstracts

05.50 64.60C 75.10H

Critical properties of the d-dinlensional

Ising n1odeI front

_a

variational nlethod

B. Bonnier and M.

Hontebeyrie

Laboratoire de

Physique Thborique (*),

^Universitk de Bordeaux I, Rue du Solarium, F-33175

Gradignan

Cedex, France

(Received

9 October 1990,

accepted

5 December

1990)

Rksumk. on considdre _un

dbveloppement perturbatif

de

l'bnergie

libre du moddle

d'Ising

_en

champ magndtique

_{constant, en} dimension arbitraire d,

qui dbpend

I tout ordre d'un

paramdtre

alors que le moddle n'en

dkpend

_pas. Ce

paramdtre

est utilisk variationnellement pour accdldrer la convergence des sbries de haute

tempbrature

de diverses observables. La mkthode,

particulidrement

efficace pour des dimensions infbrieures I d

= 2, fournit de bonnes estimations des exposants

critiques

3 et _v.

Abstract. We derive _a

perturbative expansion

for the free _energy of the d-dimensional

ising

model in _a constant

magnetic

field which, at finite order, depends upon a parameter ^{A. This} series is

expected

to

improve

the standard

high-temperature expansion

_as the model is

independant

of A,

which is used _{as a} variational parameter. The

improvement

is

particularly significant

_on the range d~ 2, as shown

by

the

computation

of the exponents 3 and _v.

1. Introduction.

Perturbative

expansions

such _as

high-temperature

series _are

frequently

used for

exploring

the critical

properties

of

spin

_systems,

especially

when the

dimensionality

_or the

symetries

of the model render its numerical simulation difficult. Rather

long

series _are,

however, generaly

needed to achieve this

goal.

On the other

hand,

various tricks

[1-3]

have been found to accelerate the convergence of

perturbative expansions

such _as those encountered with the

eigenvalues

of

quantum

Hamiltonians. In this work _we show

how,

in the

spirit

of these

methods,

_{one can} derive

high-temperature expansions

where _a variational

parameter

has been inserted in order to stabilize the results of the

analysis

at moderate order. The method is not restricted to the

Ising

model that _we consider here in

arbitrary

dimension

d,

for definiteness and in order to confront _{exact or} well established

predictions.

As _our

principal result,

_we find that the critical parameters can be

reasonably

estimated from the 8th _or 9th order

high-temperature

series _even for dimensionalities d smaller than

2,

which _are very difficult to

predict

within the standard

analyses.

This is

interesting,

since this range,

already

(*)

Unitk Associke _au CNRS, U-A. 764.

332 JOURNAL DE

PHYSIQUE

I M 3

explored

within the

e-expansion

method in reference

[4],

^is relevant for the

knowledge

of the

Ising

model in

non-integer (fractal)

dimensions

[5],

or to compare it with models

[6] suspected

to be in the

Ising

cla55 _a5 d

approaches

I.

Our method15 based _on the

partition

function Z defined

by

Z

=

jj

_exp

(tp ^jj

^«~

«~

^{+ h}

jj ^~j (I)

«=±i ,,j

where the

magnetic

field h is written _as h

= H ₊

(I t),

and _we then

expand

Z in powers of the variable _t, I-e-

Z =

jj ^~ jj (p ^jj

^«~ «~ ^A

jj

_«~ ^~

^e~~~

^{~ ~'~'}

(2)

r~o ^~' _{« >,j i}

The

physical

_{case we are} interested in

corresponds

to the value _t

=

I,

where the inverse

temperature

and the

magnetic

field take the values

p

and

H, respectively.

The true behavior of the model is then

obviously independant

of the

parameter

A, but the

expansion (2)

when

truncated at any finite

order,

does have _some functional

dependance

_on A, which _can be used

as a variational parameter. In

addition,

_we observe that the series needed _are

easily

derived from the

knowledge

of the

high-temperature

series of the free energy in _a constant

magnetic field,

which _are

already

available in the literature.

The

plan

of this work is thus to derive

explicitly

these series in the next

section,

where _we show how

using stability

conditions to determine A

permits

_u5 to

distinguish

values of

p

which converge towards the critical

coupling.

The third section 15 devoted to the determination of two

independant exponents

like and _v,

through

the critical

magnetisation

for _&, and

through

an

optimized

ratio

analysis

for _v.

2. Variational estimates of the critical

temperatwe.

The

high-temperature expansion

of the free energy

E(p, h)

of the

Ising

model reads

E( p,

h

)

=

In

(2

cosh

(h

₊

~j $

~j a( ^tanh~

^~

(h ) (3)

p~i

where the coefficients

a(

can be obtained _up to p = 9 from reference

[7]

as functions of the

dimensionality

d of the

hypercubical lattice,

the calculations

being

carried out with the

help

of the

symbolic manipulation

_program MAPLE. Derivatives of E with respect to h at h

= 0 generate observables like the

magnetisation

_or the

magnetic susceptibility.

We thus consider _X^~

= %~

E/%h

_{~, L m}

0,

and obtain their

t-expansion

from

equation (3) by making

^the substitutions

p

-

tp,

h

- H+ At. These read at order N:

x

((p, _H,

=

~o ~j

^t~

Efi(p, _{H, ) (4)}

with

Efi(p, _H,

_A

₎

=

~~

~~

%[+^{~ ln}

(2

cosh

(H

+

))

₊

r.

~

pP

P

(

~

)r

^p

+

[~ v ^{[~ al} I

p )~

~~+^{~ ~}

tanh~

~

~H

₊ A

~5)

Equivalently, collecting

_powers of

p,

one can write for the observables at t=

I,

H

= 0 the

expansions

Xl(P> ₎

"

I _P~AI,p(A)

(6)

where

A

(,

o

(A )

=

~j ~~j ^~

^{%(+ ~ In}

⁽²

^cosh

^{(A ) (7)}

and

j P N p ~ ~r

A

(

~

(A )

= ~

~j a( _~j

, %[+^~

tanh~

~

(A (8)

P' q=o r=o ~'

in such _a way that

they

reduce to the usual

high-temperature expansions

when A

=

0,

and _are

independant

of A _a5 N goes ^to

infinity.

At finite order N _we treat _{a5 a} variational

parameter

and search for values

fulfilling

the

stationarity

conditions %i

x((p,

_A

₎

=

0.

We

begin

with the _cases L _even, for which A

= 0 is

always

_a solution. In addition to this trivial

value, pairs

of

opposite

solutions _± A

(p )

can appear

depending

_on the value of

p.

To

see this it is convenient to

expand %iX~~

^{in the}

vicinity

of A

=

0,

I.e.

~~

pN

^N

~ ~~

~A~'~ ~~'

~

~ ~

_{~q ~A} ^~

^~~~h~~

~ q=>

~i-

i

); Z al

^'

~l

^~ +~

tanh~

~ A ₊ °

(A

~)

(9)

~

=

and to observe that

%(^{~ + '}

tanh~

~ A

=

C)

tanh A ₊ O

(tanh~

A

( lo)

%(~^+~

tanh~~

_A

=

C)

₊ O

(tanh~

A

(I I)

where the constants

C)

vanish if L

< q

I,

and otherwise _are

equal

to

~~

"

~~~~

^~

~~~~~~

~

Ii

o

~12)

Using

the relations

(10-12)

and the definition

~~T~ _"

~~Tfi~

^~_l

(°)/~~Tfii~ ~(°) _(l~)

the

expansion (9)

_can be written

hich

shows the apparition,

^as soon as

pm p(~, of

a

pair of

±

_A(p) iving for _the which, according to _quation (6)

corresponds to L

=

_I and is an odd unction of A.

Just

model,

the

sequences (fl(~) are

thus

^dentified _with ^timates

of

the _critical

pc,

334 JOURNAL DE

PHYSIQUE

I M 3

coefficients

A()(~(0)

_{are the usual} coefficients of the

high-temperature expansions,

I-e-

N

X~T~~~(fl,

~

"

0

=

~ fl~ _A~T~p~~(0)

=

(flc fl

^~~~~~

('~)

p=0

and

thus,

for N and p

large, A((~+~(0) PIP p~~~*~~

^' from which _one obtains

fl(~= fl~(1-

^~~~~~

⁽¹⁶⁾

N

While it is

interesting

to observe that the

stationarity

conditions

diX~~(fl,

A

)

=

0 select sequences of

couplings fl(~ converging

towards the true critical _{one, one} must admit

that, being

interested to locate it

precisely,

_we learn

nothing

_more than the ratio tests

already

Table I.

Sequences fl ( ~lirst line)

and

PI (second line)

_as N _goes

from

I _to 9.

(Stars

indicate the absence

of

real

solutions).

Each column

correspondi

to a value d

of

the dimension between 1.25 and 4 and _an estimate

of fl~

is

given

^last line

from

_a

comparison of

the _{two sequences or an} I

IN extrapolation of

the best _one, when it appears

feasible (d

~ l.65

).

N~d

1.250 1.375 1.500 1.650 1.750 1.875 2 3 4

0.400 0.364 0.333 0.303 0.286 0.267 0.250 0,167 0,125

0.730 0.664 0.608 0.553 0.521 0.487 0.456 0.304 0.228

2 0.667 0.571 0.500 0.435 0.400 0.364 0.333 0.200 0.143

0.816 0.719 0.643 0.570 0.529 0.486 0.450 0.281 0.204

3 0.783 0.641 0.545 0.464 0.422 0.380 0.346 0.203 0.144

0.856 0.738 0.649 0.566 0.522 0.475 0.436 0.264 0.190

4 1.022 0.793 0.647 0.531 0.475 0.420 0.377 0.210 0.146

0.916 0.779 0.675 0.581 0.531 0.479 0.437 0.256 0.182

5 1.297 0.900 0.697 0.554 0.490 0.429 0.382 0.210 0.147

0.954 0.806 0.692 0.589 0.535 0.480 0.435 0.250 0.177

6 0.490 0.608 0.606 0.541 0.493 0.439 0.393 0.213 0.147

1.003 0.821 0.699 0.593 0.537 0.481 0.435 0.246 0.173

7 0.509 0.567 0.587 0.542 0.497 0.444 0.397 0.214 0.148

***** 0.932 0.698 0.594 0.539 0.481 0.434 0.243 0.170

8 0.20 0.47 3 603 0.756 0.561 0.461 0.404 0.215 0.148

***** ***** ***** 0.606 0.546 0.484 0.434 0.241 0.168

9 0.308 0.213 0.000 1.248 0.611 0.470 0.407 0.215 0.148

***** ***** ***** 0.613 0.554 0.486 0.434 0.239 0.166

flc I.,

1.3

0.9,

1-1 0.75

0.61,

0.63 0.56 0.49 0.44 0.22 0.149

performed

in the framework of the

high-temperature analyses.

We

have, however,

at _our

disposal

the

stationary

^conditions on L odd. As _we cannot treat

analytically

these

constraints,

we restrict their

study

to the

magnetisation (L

=

I)

which is

expected

to

give

the best sequence,

just

_as the best sequence for L _even is

given by

the ratios of the

magnetic susceptibility,

associated to the

stability

of the free energy itself L

=

0. It then appears in the

case L

= I that at small values of

fl

there _are

always

_pure

imaginary

solutions A and when

fl

increases above _some threshold

p(,

_{one can} find in addition

pairs

of real solutions

±

(p ).

As in the

previous

_case, this _sequence

(p ()

indicates critical

points

and is fisted with the sequence

(p()

_{in the table I} for various dimensionalities. From

equation (16)

p(

~

p~ (1-

^~

_),

where _y is the

magnetic susceptibility exponent,

^known to decrease

N

from _oJ to I _as d increases from I to 4

(Ref. [3]).

This I

IN

behavior is

apparent

^{at low order}

only

when d~2 and then the best sequence

corresponds

to L

=

0,

but when d<2 _a

comparison

of the two sequences is needed to

give

_an ^{estimate of}

p~(d).

3. Variational estimates of the critical exponents.

The

previous

results

suggest

that _we

investigate

further the

magnetisation,

which in presence of _a small

magnetic

field H must behave _a5

H'M

_{at the critical} temperature. ^The present

approach gives

_us the

opportunity

to evaluate the exponent

through

the

expression

N

3

(H,

A

)

= =

jj

t~

3~(H,

A

(17)

H

dH

^In _X

(flc, H,

h

)

, o t i

where the series

appearing

in Def.

(17)

is derived from the

expansion (4).

We then represent 3

(H,

A

) by

the Padb

approximants

formed from this series in t,

subsequently

set

equal

to its

physical

value t

= I

and,

since N _«

9,

_we focus _on the

[4, 4], [3, 5], [5, 3], [4, 5]

and

[5, 4]

approximants.

It appears that when H is

large,

H»

H~,

^{it is} _easy to find the values of A, A

= A

(H),

which stabilize all the

approximants

around _{a common} value for

3(H,

A

(H)) expected

to be close to the exact one, as for the _case d

= I that _we have checked. We then

decrease H in order to find _a

region

H=

H~

where

3(H, (H)) displays

_a minimal

dependance

with

H,

and _{we assume} that the

exponent

^{3 is}

given by

3

=

3

(H~, (H~) ).

If _we decrease further

H,

_we find _a

spurious

increase of

3(H, (H))

and for smaller fields _no

prediction

_can be done. This behavior is illustrated _on the

example

shown in

figure I,

where

d=2 and _3~~~~~

=15,

and where _we estimate 3

=15±0.2,

within _an effective field

H~

_~ ^{0.06 and values} of A~ = A

(H~)

in the range A~ ~

0.44 0.47. More

generally,

the results

predicted by

this method for various dimensions d below 2 _are listed in the table II.

(Above

d

=

2,

_our results agree with the values

already known.)

It appears that the determination of 3 is very sensitive to the

precise

location of

p~,

but when _one considers the

exponent

~ as to

compare with its

prediction

_~~z

given

in reference

[4],

one finds _more accurate values

always

in agreement ^with _~~z.

We _now turn to the evaluation of _an other exponent, ^and to eliminate the uncertainties linked to the determination of the critical

point p~,

we use a « critical

point

renormalization _»

method

[7, 8]

that _we recall below.

Suppose

_we have two functions

F~(p ),

k

=

1, 2,

with _a

common smaller

singularity p~

Fk(P

"

z ^P~~l Ck(' P/Pc)~~~

k

"

',

2

('8)

then the function

R(x)

defined

by

the series

R(x)

=

jjx~a)Ia~ behaves,

for _{x near}

I,

as

r

336 JOURNAL DE

PHYSIQUE

I M 3

(1- x)~~~~'

^~' Thus _{a, a~ +} I is

given by

^the value of

L(x)

=

(I x)

d~

log R(x)

at

x ₌ I.

Choosing Fj

₌

x((p,

A and

F~

=

xl _(p,

_A

_)]~,

_we have

a j a~ = y~ 2 y~ =

vd,

and thus

v =

(L(x

=

I) I)/d. (19)

0.46

~<S

Ii

Fig.

orrespond to

[4,

5] and [5, 4] Padb approximants of 3 [H, A (H)]. Numbers along _these

the

values

A _(H) here these ave extrema

with

espectto

the

_parameter A.

Table II. Estimates

of

the exponent 3

for

various values

of

the dimension d between I and 2

as

function of

_an assumed value

p~ for

the critical temperature. ^The

corresponding

values

of

_~,

~ =

(2

+ d ₊

(2 d)

3

)/(3

₊

1),

are

compared

to the

predictions

_~~z

of reference [4].

d

p~ H~

_A~ 3 _~ ~~z

1.250 0,15 0.9 30-33 0.82-0.83

1.15 0.20 90-100 0.775-0.777 0.30-1.00

1.30 0.25 1.2 280-320 0.758-0.759

1.375 0.9 0.15 1.04 50-55 0.674-0.679 0.30-0.80

1-1 0.15 1.26 230-260 0.635-0.637

1.500 0.70 0.20 22-28 0.603-0.631 0.35-0.65

0.75 0.20 41-44 0.567-0.571

1.650 0.61 0.15 0.68 20-24 0.482-0.507 0.30-0.50

0.63 0,15 0.68 25-32 0.450-0.477

1.750 0.56 0.13 0.64 19-23 0.396-0.425 0.30-0.40

1.875 0.49 0.13 0.57 18-19 0.312-0.322 0.27-0.33

2 0.44 0.06 0.45 14.8-15.2 0.247-0.253 0.25

Our

procedure

to evaluate

L(x

=

I)

is to form Padb

approximants

to

L(x), starting

from

equation (6)

for

xl

and

xl,

and to search the values of which stabifize _v

given by equation (19).

The series in the variable _x

giving L(x)

is known up ^to the 8th order and

thus,

to

simplify

the

analysis,

_we select _a set of

6th,

^{7th and} 8th order

approximants ([4/2],[3/3], [2/4], [3/4], [4/3], [3/5], [4/4], [5/3])

and fix A _as to minimize the

dispersion

of the values of

v

given by

this set. This crude choice does _not

significantly improve

the determination of _v for d

~ 2

(for example

at d

=

2,

0 and 0.991 _{w v w}

1.014).

At lower dimensions

however,

_as

shown in the table

III,

_pure

imaginary

values of allows to obtain _a set of coherent

values,

task _we have found

impossible

within _a standard Pad]

analysis.

In

conclusion,

it appears that the methods _{we use are} able to

compete,

^{for d<}

2,

with the

sophisticated

summations of the

e-expansion performed

in reference

[4],

^and _our results _are in full agreement ^{with this work. At}

higher dimensions,

the

improvement

is _{not so}

striking,

but _we want to

point

out that the

magnetisation

_seems to be the best observable to

analyse

within the

present approach.

It _can be thus of _some

help

^for models where the order of the transition is difficult _to assert.

Finally,

_we want to

point

out that the usefulness of such variational series has been also

recently demonstrated,

in _a

slightly

different context and

through

other

methods,

in the work of reference

[9].

Table III. Predicted ranges

of

values

for

the exponent v

for

various dimensions

from

the

approximants of

section

3,

when is

fixed

_at the

complex

value

given

in column

Ali.

The numbers in column v~z are taken

from reference [4].

d

Ali

_{v v~z}

1.250 0.035 1.7-3.4 1.5-4.5

1.375 0.016 1.6-2.9 1.6-2.6

1.500 0.016 1.49-1.84 1.45-1.85

1.650 0.012 1.27-1.38 1.30-1.44

1.750 0.009 1.18-1.26 1.20-1.26

1.875 0.002 1.ll-1.13 1.09-1.ll

Acknowledgements.

We _are

grateful

to Andrb Neveu for

having suggested

this

investigation

and for

stimulating

conversations with Gerard Mennessier and other

colleagues

in

Montpellier.

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