On the one-dimensional ising model with arbitrary spin

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On the one-dimensional ising model with arbitrary spin

R.L. Bowden, D.M. Kaplan

To cite this version:

R.L. Bowden, D.M. Kaplan. On the one-dimensional ising model with arbitrary spin. Journal de

Physique, 1976, 37 (7-8), pp.803-811. �10.1051/jphys:01976003707-8080300�. �jpa-00208476�

ON THE ONE-DIMENSIONAL ISING MODEL WITH ARBITRARY SPIN (*)

R. L. BOWDEN and D. M. KAPLAN

Department

^of

Physics, Virginia Polytechnic

^{Institute and State}

University, Blacksburg, Virginia 24061,

^U.S.A.

(Reçu

^{le 8 décembre}

1975, accepté

^{le 23 mars}

1976)

Résumé. ²⁰¹⁴ Comme illustration d’un formalisme quasi classique ^de spin développé ^{dans un autre} article, ^une nouvelle méthode pour résoudre un modèle

d’Ising

unidimensionnel est présentée.

La méthode

employée

^{est une extension aux} systèmes ^de spin de la méthode de

Wigner

familière dans les calculs qui comprennent ^la position ^et

l’impulsion.

^La fonction de partition, la fonction de corréla- tion de paires ^{et la}

susceptibilité

pour champ ^nul pour le modèle d’Ising unidimensionnel de spin

arbitraire sont exprimées ^au moyen de fonctions propres ^et de valeurs propres d’une certaine équation intégrale dont le noyau

approche

le noyau classique tandis que le spin approche ^l’infini.

Abstract. ²⁰¹⁴ As an illustration of a

quasiclassical

spin ^formalism developed ^{in a recent} paper ^{a new} method for solving the one-dimensional Ising ^{model is} presented. ^The method used is an extension to

spin systems ^{of the} Wigner method familiar in computations involving position ^{and momentum.}

The partition function, pair correlation function and the zero-field susceptibility for the one-dimen- sional Ising ^model

with

arbitrary spin ^{are shown to be} expressed ^{in terms of the} eigenfunctions ^and

eigenvalues

^{of a certain} integral equation whose kernel approaches the classical kernel as the spin approaches infinity.

LE JOURNAL DE PHYSIQUE

Classification Physics ^Abstracts

0.230

1. Introduction. ^- In a recent paper

[ 1 ],

^a

quasi-

classical

spin

^{formalism was}

developed

^for

spin algebras

characteristic of

Ising

models. Since such

quasi-classical spin

formalisms have received recent attention

[2],

^we illustrate here its use in a rather detailed

analysis

of the one-dimensional

Ising

^model

with

arbitrary spin

^{S. It is} well known that such a

problem

^{can be set} up ^{as a} matrix

problem

^{in the same}

way Kramers and Wannier

[3]

^set up the S

= !

^pro-

blem. For

example, Suzuki, Tsujiyama

^{and Katsura}

[4]

set up the

problem

^{in this manner, and}

using

^a per- turbation

technique

^{and an}

implicit

differentiation

technique,

^obtained

explicit

results for

S = 1, 3/2.

Furthermore, they

showed that the matrix

problem

can be reduced to order of the

greatest integer

^not

exceeding

^{S + 1.}

Thus,

^it should be

expected

^that

their

techniques

^{can be used to obtain}

explicit

^results

(*) ^This paper ^was supported ⁱⁿ part by ^the National Science Foundation, Grant # GK-35-903.

for S _ y. However,

^it appears that it would be a very formidable task to use their

techniques

^{to obtain}

explicit

results for S >

y.

^On the other

hand, Thompson [5]

^{solved the classical}

Ising

^{model in}

which S > oo in terms of the

eigenfunctions

^and

eigenvalues

^{of a certain}

integral equation

for which the

exponential

of the classical Hamiltonian is the kernel.

In this _paper we use a

recently

formulated computa- tional

technique [1, ^6, 7]

^{in which a}

mapping

^of

spin

matrices onto

polynomials

allows the

theory

^of

spin

to be

expressed

^{in the}

language

of continuous

variables.

This

technique

^{is an} extension of the

Wigner

^method

[7- 10]

^{familiar in}

computations involving position

^and

momentum. In

particular,

^we

apply

^the

Wigner

’

method

appropriate

for matrix

algebras

which contain

only commuting

^matrices characteristic of

Ising

^models

as outlined in reference

[ 1 ].

^Reference

[ 1 ] verifies

^the

mapping

rules for a

spin

^S

algebra

^formed

by spin operators Si, S2,

^...,

SN, identity

operator

I,

^and

arbitrary

functions of the

spin operators

^{for which}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01976003707-8080300

804

we use the notation

A(Sz,

^...,

SN).

^The

mappings

for

Sf

^{and I are}

given by (we

^denote

mappings by

arrows

--+) Sit SOi

^--_

[Si ]w (see

^eq.

(4))

^and

I --+ 1

--- Iw,

where S is the

spin

quantum ^number and

Qi

^{is a} continuous variable with _range

( - 1,

⁺

1).

We call the function into which the operator ^{has been}

mapped

^the

Wigner equivalent

function. Thus

[Si ]w

and

Iw

^{are the}

Wigner equivalent

functions for _opera- tors

Sf

^{and I. To} obtain the

mapping

of functions of

Si

^a constructive

principle

^can be stated as

follows : If

Aw(f2l,

^...,

QN)

^and

Bw(Ql,

^...,

QN)

^{are the}

Wigner equivalents

^{of two}

arbitrary

operators

A(S’,

^...,

SN)

^and

B(S1, ..., SÑ),

^{then the}

Wigner equivalent

^{of the}

product A(Sz,

^...,

SN) B(Sz,

^...,

SN)

is obtained

by

^{the rule}

(see

eq.

(6))

(A B)w (f2 1, - -, QN)

⁼

A,,,(Q 1, - - -; ON) GBw(D1, ON)

where G is a certain

left-right

differential operator

(see

eqs.

(7)

^and

(8)).

^{The trace of an} operator goes

over into an

integral

^{of its}

Wigner equivalent (see

eq.

(3)).

^We find that the

partition function, pair

correlation

function,

and the zero-field

susceptibility

for the one-dimensional

Ising

^{model with}

arbitrary spin

^{can be}

expressed

^{in terms of the}

eigenfunctions

and

eigenvalues

^{of a certain}

integral equation

^whose

kernal

approaches

^the classical kernel as S - XJ.

The results obtained here are valid for all values of S

including

the classical limit

and,

^of course, for finite S the results obtained here contain those of

Suzuki, Tsujigama

^{and Katsura.}

We

proceed

^{as follows : In section 2 we obtain the}

above mentioned

integral equation by applying

^the

Wigner

^{method to} get ^the

partition

^{function. We}

study

the

properties

^{of this}

integral equation

^{in section 3.}

In section 4 we obtain

expression

^{for the}

pair

^correla-

tion function and the zero-field

susceptibility. Finally,

in section 5 we look at the classical limit.

2. Partition function. ^- Consider a one-dimen- sional lattice with an

Ising particle

^of

spin

^{S at each}

lattice

site,

^{labeled i =}

1,

^{..., N.} The Hamiltonian of this system ^is

given by

where Sf

^{is the} associated

spin

^matrix of the ith

particle

^{and J measures the}

magnitude

^{of the}

exchange

interaction. The

partition

^function

ZN

^is

given by

where

fl

⁼

1/kT.

^The

Wigner

^method

[1] provides

^a

technique

for evaluation of the trace of _any function of the

spin

operators,

A(S’,

^...,

SN), by integration

^of

its

Wigner equivalent,

i.e.,

where the

Wigner equivalent

^of

Sf

^is

given by

Thus,

^{we have}

The

Wigner equivalent

^of

products

of functions can

be obtained

by

^a Groenewold rule

[1]

where the operator ^{G is}

given by

and

Gi

^{is the}

right-left

differential operator

We can then write

Since

[ePJSNS1]w

^{is a} function of

ON

^and

D1 only,

^we

have

Now

integrating by

parts ^on

QN,

^{we obtain}

where [1] ]

Repeating

^this

procedure

^{N - 1}

times,

^{we find}

where

K(N)(Q, Q)

^{is the Nth} iterate of the kernel

with

We

verify

^{in the next} section that

where

the A,

^{are the}

eigenvalues

^{of the}

integral equation

Therefore,

^the

partition

^{function is}

given by

and for a

long

^{chain the}

partition

^function per lattice

site, Z,

^{is then}

given by

where

Ao

^{is the}

largest eigenvalue.

3.

Integral equation.

^{- We now look at the}

eigen-

values and

eigenfunctions

^{of the}

integral

eq.

(16).

It is however convenient to first examine the

adjoint integral equation

Integrating by

parts ^on

02 in

eq.

(19),

^{we obtain}

.

from which we can write

Therefore,

^if

cpT

^{is an}

eigenfunction

^with

eigenvalues A, satisfying

^the

adjoint

eq.

(19)

^then

is an

eigenfunction satisfying

eq.

(16)

^{also with}

eigenvalue À"

To further

study

^the

eigenvalues

^and

eigenfunctions

of _eqs.

(16)

^and

(19)

^{we need a}

representation

^of

K(Ol, Q2)-

^One

representation

^{can be obtained}

by noting

^that

K(Q 1, (2)

^is

given by

^the

equation [1 ]

where the

operator

is defined

by

Now as can

easily

^be

^verified,

^the

operator

^{has the}

eigenfunctions

with

eigenvalues

ⁱ

where 2S is S1

^a binomial coefficient.

Using

^the

representations

we can write

from which we obtain

Thus we can write from _eq. (

Let us now

expand CPt

^and CP, in

Legendre poly-

nomials,

806

and note that

[1]

where

Therefore from _eqs.

(20)

^and

(29),

^{we can write}

Multiplying by Pm(Q1)

^and

integrating

^on

Ql

^over

(- 1,

⁺

1),

^{we obtain}

where the column vector A (I) has the elements

A§,

n ⁼

0, ..., 2 S;

the transfer matrix V has the ele- ments

[11 ]

the matrix F has the elements

with

3F2

^a

generalized hypergeometric

function of unit argument; ^the

diagonal

^{matrices M and G have}

elements

[(2 m

⁺

1)/(2 S

⁺

1)] ^bi,,, and gn bin,

^res-

pectively.

^Here

bi,,

is the Kronecker delta function and t

denotes the transpose. ^{We show in the}

appendix

^that

Hence if

C(l)

is the column vector

given by

then

From _eqs.

(22), (31)

^and

(32),

^we also obtain

where the column vector B (l) has elements

B.(’),

n ⁼

0,

^{..., 2 S.} Therefore the

eigenvalues

of the inte-

gral

eqs.

(16)

^and

(19)

^{are the same as the}

symmetric

matrix V.

Furthermore,

^the

eigenfunctions

^and

cpT

^can be constructed from

knowledge

^{of the}

eigen-

vectors of V.

It is

easily

verified that

K(Qi, (2)

⁼

x(- 01, - (2),

so that _T, and _T, can be constructed with definite

parity.

To further

classify

^these

eigenfunctions,

^consi-

der the

eigenvalues

^and

eigenvectors

of the matrix V.

We will first consider the

ferromagnetic

interaction

case J > 0. If we let

and

the elements of V can be written as

the matrix V is a

generalized

Vandemonde matrix and thus is an

oscillating

^{matrix with the}

following

^two

important properties [12].

^{First the}

eigenvalues

^{of V}

are all

positive

^and

distinct;

^we order them such that

Second if the

eigenvectors ^C(’)

of V have the elements

C,(,’),

^the sequence

{Cf}),

^...,

C2s },

^I

fixed,

^has

exactly

^I

changes

^of

sign.

On the other

hand, Suzuki, Tsujiyama

^and

Katsura

[4] (see

also Dobson

[13])

indicated that V

can be transformed into block

diagonal

^{form with}

a

similarity

transformation. To be

specific,

^define

the

partitioned

^matrix

or

and let

where if

S ±

are

integers

^{such that}

then _y is a column vector of

dimension S -,

^with

elements

equal

^to

unity,

^{I is the S - x S - unit}

matrix and I’ is the S - x S - matrix with elements

I q p = ðs- -q,p’ q,p

⁼

0, ..., S - -

^{1. We then find} where

with

or

and the S - ^x S - matrices V + and V - have elements

Thus the

eigenvalues

^{of V are the}

eigenvalues

^{of U +}

plus

^the

eigenvalues

^{of V -.}

Furthermore,

^{there are} S + and S -

independent eigenvectors

^{of U in the}

forms

respectively.

^{Now with} eq.

(51)

and the fact that the sequence

{ C(1),

^...,

C(1) }

^has

exactly

¹

changes

^of

sign,

we find that the

eigenvectors ^C(l)

^must have the forms

and

We now turn to the

antiferromagnetic

interaction

case J 0. First note that if V is defined

by

eq.

(36),

then

where here the matrix I’ has

elements 62S-q,p,

p, q ⁼

0,

^{..., 2 S.} Furthermore from _eq.

(53)

^{we find}

Therefore if

CO) and A,,

^{1 =}

0,

^{..., 2 S are}

eigenvectors

and

eigenvalues

^of

V(J

>

0),

^then

^C(l)

^and

(- 1 )l À,

are

eigenvectors

^and

eigenvalues

^of

V(J 0)

^since

Thus,

eq.

(45)

^{can be}

generalized

^{for real J to}

It is

easily

verified that

Fqn

⁼

(- 1)n F2s - q,n’

^There-

fore from _eqs.

(39), (41)

^and

(50)

^{we find}

It follows then from _eqs.

(30)

^and

(31)

^that

and

The functions

CPI(Q)

^are

linearly independent

^{since if}

then

which

implies each fli

^{= 0} since the determinant of F-1 does not vanish and the

eigenvectors ^C(l)

^are

also

linearly independent.

The functions

CPI(Q)

^form

a basis for

polynomial

functions of

degree

^{2 S. Thus}

we obtain the bilinear

expansion

where we have assumed the

orthogonal

’functions _CP, and

pT

have been normalized so that

Substitution

of eq. (59)

^into eq.

(12)

^leads

immediately

to eq.

(15).

4. Correlation functions and

susceptibility.

^{- The}

pair

correlation and zero-field

susceptibility

^{can also be}

obtained

by

^the

Wigner

^{method. The}

pair

correlation function can be written as

The

Wigner equivalent

^{statement is}

808

Integrating by

parts ^on

QN,

^{we obtain}

Repeating

^this

procedure,

^{we find as an} intermediate step

Continuing

^this

procedure,

^we

finally

^obtain

If we now substitute from _eq.

(63),

^{we obtain}

where

For fixed _r, we find

From the fluctuation

relation,

^the

susceptibility

^{in zero field for N}

particles

^is

(with

^each

spin having

^unit

magnetic moment)

Substituting

^from eq.

(70),

^we find after a little

algebra

Therefore

Noting

^that

we can write

where we have assumed that

according

^to eqs.

(22), (32)

^and

(64)

^that

Eqs. (72)

^and

(75)

^{are exact}

expressions

^{for the}

pair

correlation function _p and the zero-field

susceptibility

x for

arbitrary

^s.

They,

^{of course, contain the}

eigen- values A,

^{and the}

eigenvectors

^B(l). Since the matrix V

can be transformed into block

diagonal

^{form so that}

each

resulting

matrix is of order of the greatest

integer

^not

exceeding S

⁺

1,

^{and since one can in}

general

^{solve a} fourth order

equation,

^these

equations

can be used to obtain

explicit

^results

for S 7/2.

The details are omitted here.

However,

^{with the aid of}

high-speed

computers, the matrix

problem

^{can be}

numerically

solved and _p and _x calculated from eqs.

(72)

^and

(75).

^This

procedure

^was followed for

calculating Z

and the results are

presented

^{in the next}

section.

5. Classical limit. ^- For very

large

^{values of S,}

these results are

approximated,

^{at least for}

high

temperature,

by

the classical

Ising

^model

[5]. Moreover,

the

Wigner

method used here leads

naturally

^{to the}

classical model.

Perhaps,

^{the easiest} way ^{to see} this is to note that if we

prescribe

^that

^{JS2 =}

^{J* = cons-}

tant oo for all

S,

^we get ^from eq.

(23)

Hence,

^{we have}

and

where

cpi(Q)

^and

Ài

^{are the}

eigenfunctions

^and

eigenvalues

^{of the}

integral equation

This is

just

^the

integral equation

^used

by Thompson [5]

to

analyze

the classical

Ising

model and for which the

eigenfunctions

^are the oblate

spheroidal

^{wave func-}

tions

[14].

The

Wigner

^{method can} also be used to obtain quantum corrections to the classical model. To see

this,

^{let the}

operators X

^{and Y} be defined as

and

.

We can then write from the Baker-Hausdorff for- mula

[15]

FIG. 1. ^- Inverse susceptibility ^for ferromagnetic interaction.

The ordinate and abscissa denote PS(S + 1 )/x ^and l/BJ*, ^res- pectively.

FIG. 2. ^- Susceptibility ^for antiferromagnetic interaction. The ordinate and the abscissa denote x/flS(S + 1) ^and 1/BJ*, ^res-

pectively.

where the operator ^{W is}

given by

with

[X, Y] = [XY - YX].

^{Now the} operator ^{W can} be

expressed,

^{at least in}

principle,

^{as a series of} ope- rators

Wn

^{such that}

Thus quantum corrections to the classical

Ising

^model

can be

systematically

calculated

by perturbation expansions

^{in the}

quantity 1 /S.

^For

example,

^{to order}

1/S,

^{we find}

810

where

A standard first order

perturbation

calculation then yields

and

where

Substitution of _eqs.

(90)

^and

(91)

^{into the}

expressions

of the

preceding

sections will

yield

^the

partition function, pair

correlation function and zero-field ^.

susceptibility

correct to order

1/S. Higher

^order

corrections can be obtained in a similar manner.

Graphical displays

^{of the}

approach

^{to the}

classical

limit are shown in

figures

^{1 and 2. The}

graphs

^shown

in the

figures

^{are some} of the results of

numerically analyzing

^the

equations

^{derived in the}

preceding

section for different values of the

spin S keeping

^J*

constant. The

susceptibilities

^{shown in}

figures

^{1 and 2}

were calculated

by numerically solving

^{for the}

eigen-

values and

eigenvectors

of the matrix V and

applying

eqs.

(41)

^and

(75).

Appendix.

^{- In order to} prove eq.

(38)

consider the

integral equation

with the

degenerate

^kernel

The

eigenfunctions of eq. (A .1)

^{are the}

Legendre polynomials

^{and the}

eigenvalues

^are

given by

We prove this last statement

by expanding Ks(x, y)

^{as a bilinear}

expansion

^of

Legendre polynomials.

^{In order to}

obtain this

expansion,

^{we use the}

representation

^{for the}

Legendre polynomial [16]

io write

Two other

representations

^of

Ks(x, y)

^can be obtained

by considering

^the

expansion (see

^for

^example

^ref.

^[16],

p.

185)

Letting

and

using

the addition theorem

where

Plm(x)

is the associated

Legendre ^function,

^{we find}

However,

^from

Laplace’s

^first

integral

^{for the}

Legendre polynomial,

^{we have}

Thus from _eqs.

(A. 9)

^and

(A. 10),

^we get

Now

using

eq.

(A. 5)

^{and the}

orthogonality

^{of the}

Legendre polynomials,

^{we find}

If we use

Ks(x, y)

^from eq.

(A. 2)

in eq.

(A. 12), multiply

eq.

(A. 12) by Pk( y)

^and

integrate

on y ^over

( - 1,

⁺

1),

we obtain the

interesting (but

^{difficult to}

directly prove)

^result

Eq. (38)

follows almost

immediately.

References

[1] BOWDEN, ^{R. L.} and KAPLAN, ^D. M., Quasi-classical ^Formalism for Ising ^Model Algebras. Submitted for publication ^1976.

[2] CHANG, ^Y. I., SUMMERFIELD, ^{G. C. and} KAPLAN, D. M., Phys.

Rev. 3 (1971) 3052; CHANG, ^{Y. I. and} SUMMERFIELD, G. C.

B 4 (1971) 4023; EBARA, ^{K. and} TANAKA, Y., ^J. Phys. ^Soc.

Japan ³⁶ (1974) ^93.

[3] KRAMERS, ^{H. A. and} WANNIER, G., Phys. ^{Rev. 60} (1941) ^252.

[4] SUZUKI, M., TSUJIYAMA, ^{B. and} KATSURA, S., J. Math. Phys. ⁸ (1967) 124.

[5] THOMPSON, ^C. J., ^J. Math. Phys. ⁹ (1968) ^241.

[6] KAPLAN, ^{D. M. and} SUMMERFIELD, G. C., Phys. ^{Rev. 187} (1969)

639.

[7] KAPLAN, ^D. M., Transport Theory ^and Statistical Physics ¹ (1971) 81.

[8] WIGNER, E., Phys. ^{Rev. 40} (1932) ^749.

[9] GROENEWOLD, ^H. J., Physica ¹² (1946) ^405.

[10] MOYAL, ^J. E., Proc. Cambridge ^{Philos. Soc. 45} (1949) ^99.

[11] This is the same matrix obtained by SUZUKI, TSUJIYAMA and KATSURA [4].

[12] GANTMACHER, ^F. R., Application of ^the Theory of ^Matrices (Interscience Publishers, Inc., ^N. Y.) ^1959.

[13] DOBSON, ^J. F., J. Math. Phys. 10 (1969) ^40.

[14] FLAMMER, C., Spheroidal ^{Wave Functions} (Stanford University Press, Stanford, California) ^1957.

[15] MAGNUS, W., Comm. Pure Appl. ^{Math. 7} (1954) ^649.

[16] RAINVILLE, ^E. D., Special ^Functions (The ^Macmillan Co.,

New York) ^1960.