On the use of different matrix products to solve finite three-dimensional Ising systems

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On the use of different matrix products to solve finite three-dimensional Ising systems

Ph. Audit

To cite this version:

Ph. Audit. On the use of different matrix products to solve finite three-dimensional Ising systems.

Journal de Physique, 1986, 47 (7), pp.1119-1126. �10.1051/jphys:019860047070111900�. �jpa-00210298�

On the use of different matrix products ^{to solve} finite three-dimensional

Ising ^systems

Ph. Audit

Laboratoire PMTM, Université Paris-Nord, ⁹³⁴³⁰ Villetaneuse, ^France

(Reçu ^{le 9 dicembre} 1985, accepté

forme définitive ^{le 24}

1986)

Résumé. 2014 On définit

produit matriciel, qui peut être utilisé

liaison avec trois autres produits

matriciels classiques, pour résoudre analytiquement ^des systèmes d’Ising ^de taille finie. A titre d’exemple,

^déter-

mine l’expression ^exacte de la fonction de partition ^{du réseau} périodique anisotrope ²

²

²

présence ^de champ, qui ^{conduit aux} expressions ^exactes de l’aimantation et de la chaleur spécifique ^de

système.

Abstract.

A

matrix product ^{is defined} and used in connection with three other well-known matrix products

in order to derive analytical solutions of finite three-dimensional Ising systems. ^As

example, ^{the exact} partition

function of the periodic anisotropic ²

²

2 lattice in a field is evaluated from which analytical expressions ^for

the magnetization ^{and the} specific ^{heat of this} system ^follow.

Classification

Physics ^Abstracts

75.40D - 05.50

1. Introductioa

The most elegant ^and straightforward solutions of one-

dimensional systems ^{of N} Ising spins ^are supplied by the well-known transfer matrix formalism [1],

where the partition function is given by ^{the trace of a} regular product of N matrices. However, ^the difficulty

of calculating ^exact eigenvalues ^{for the} large ^matrices

involved in the solution of three-dimensional systems has prevented the above result from being generalized.

Notwithstanding the need of new matrix products ^to

build the transfer matrix has been clearly ^realized by Novotny [2] ^who proposed ^{the row} and column matrix product ^for this purpose. Analogically, ^the

basic tool of the present ^{method is a new matrix} product well suited to the three-dimensional Ising problem; ^and making ^{use of this} product, ^the partition

function is obtained as the trace of a combination of Pauli matrices, ^{which can be} calculated without _any

diagonalization. ^{We will} explain the method and

apply ^{it to the} Ising 2 x 2 x 2 lattice in the _presence of a field, ^{with different} couplings in the three direc- tions of _space and assuming periodic boundary

conditions. In spite ^{of the} simplicity of the model

considered, ^the derivation of its solution is actually

non trivial, ^but quite instructive. Indeed, by ^consider- ing the derived analytical result, ^{the different roles} played by ^the dimensionality, ^the anisotropy ^{and the}

field respectively ^are strongly illuminated, ^{and the} difficulty encountered in seeking solutions for Ising

models taking ^those parameters ^more fully ^into

account can be well acknowledged ^and explained Furthermore, ^{from the} computer assisted derivation of the isotropic ^{4 x 4 x 4} Ising ^model partition

function [3], ^{we have learned much} of the difficulties encountered as the lattice size increases. In contrast, the considerable complexity arising ^{from the aniso-}

tropy ^{in the} presence of ^the field for 3d models has never, to ^our knowledge, been treated yet. ^{This has} guided ^our choice for the first model to be solved

by ^{this new} method, the solution of models with ^a much larger size will be given ^{in a} subsequent paper.

In the present paper, ^we consider the partitions

function of ^a system ^{of N} Ising spins Qjmn ^{on a} three- dimensional lattice, ^with coupling ^constants Ji, (i

^1, 2, 3), along ^{the x, y, z} directions, ^{in an external} field h

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047070111900

1120

with Ki

JJkB T ^{and H}

h/kB ^{T. In order to set}

up ^a matrix characteristic of the three-dimensional

lattice, ^{we start} by defining ^{the four 2 x 2 matrices} Pl, P2, P3 ^and Q (diagonal) ^as

and in terms of the Pauli matrices

we have

where

and we give also for future use the matrices

Thus, ^the partition ^function (1) ^can be written in the form

suitable for an evaluation using the matrix formalism to be developed ^{in the} following.

2. Matrix products.

Besides the regular product

^{denoted AB - of two}

square matrices, ^use will also be made of the direct

product ^defined by

In addition the Hadamard product ^defined by [2, 4]

will also be needed. Useful examples of this matrix

product ^are given below, taking ^{into account} equa- tions (5), (6), (7), (12) ^and (13)

with

Interesting relations link the direct and Hadamard

products ^as

which follow directly from definitions (15) ^and (16).

However, ⁱⁿ equation (14), ^the contribution of a

given spin ^{involves a summation over its six} neigh- boring spins; hence, it is necessary ^to define a fourth matrix product specially designed ^to take the three- dimensional Ising interactions into account Therefore,

we will now consider six arbitrary 2 x 2 matrices A, B, C, A’, B’, ^{C’ and one} diagonal ^{2 x 2 matrix}

D

diag (d1, d2). ^The following expression

can be regarded ^{as a} generalization ^{of the} regular

matrix product, where, ^{in contrast with the} latter,

the sum over j ⁱⁿ (23) ^is restricted to the products ^of

the first (respectively last) element in the A p B © ^C matrix row with the first (respectively last) ^element

in the A’ p ^B’ © ^C’ column and with the first (respec- tively last) diagonal element of the matrix D. By ^means

of the diagonal ^matrix

our special ^matrix product ^can be written in the

convenient form

In terms of the Pauli matrices (3), the matrix L1 reads

Using either of those expressions, equation (25) gives

3. The partition function of the anisotropic ^{2 x 2 x 2 lattice.}

To illustrate the usefulness of the matrix product defined in section 2, ^we now, consider the 2 x 2 ^x 2 Ising

model endowed with the full complexity resulting ^{from the} couplings’ anisotropy ^{and the presence of a field,}

in spite ^of its small size. Besides, the model is assumed to satisfy periodic boundary conditions, which in this

special case, means, that each spin interacts twice with each of its three neighbours, ^{so that the model} actually displays ^a full three-dimensional character.

°

The eight spins ^{of this} lattice, ^{labelled as in} figure 1, ^{can be} regarded ^as split ^{into two} layers : (61, 0’3’ U:51 (J7)

and (Q2, ^{(J 4’} (J 6’ 6s), ^{and the terms of the} product ⁱⁿ equation (14) ^{can be ordered} accordingly by using equa- tion (15), ^{so that the} partition ^function of this model can be written as

Fig. ^1.

Spins

^{the 2}

²

^{2 lattice.}

1122

After a summation over the second layer, ^{or even} spins, according ^{to the rules} (28) ^or (29), ^the following expression

is left over

which involves the first layer, ^{or odd} spins, only; and where for brevity, ^{we have set}

Taking ^{into account} the direct product definition (15), equation (31) ^{now reads}

or by rearranging matrix elements in the direct products ^{we have}

but further reduction results from using ^{the Hadamard} product (16) ^{and we obtain}

Then, taking equations (17), (22) ^and (32) into account, ^we introduce the following ^notation

and taking equation (22) ^{into account, we can} write several expressions ^{such as}

and

which yield ^the following ^relations

so that the partition ^function (34) ^is given by ^{the trace} of the Hadamard product ^{of two} matrices, which differ from one another by ^{the order of the terms} in the direct products only, ^{it is}

where the properties ^{of the trace of a direct} product, ^{as well as the} invariance of the trace under cyclic permutation

of factors have been used.

Now, application ^{of rule} (22) ^to expressions (35) gives

those matrices can be expressed ^{in terms of} the Pauli matrices from equations (16) ^and (17), and that makes the

calculation of their traces trivial; ^{we obtain} successively

1124

and

Finally, substituting equation (41) ⁱⁿ equation (38), ^we obtain the desired result

We note that the derivation of this result required ^{the use} of all four different matrix products considered in section 2 ; ^this necessity should arise in seeking the solution of _any three-dimensional Ising system, ^{in contrast} with the one-dimensional case where the use of the regular ^matrix product is sufficient.

It is straightforward ^{to derive} particular expressions ^{from the} general ^formula (42), ^{such as} for instance the zero-field partition function which is obtained by simply setting y

c3, ^g

S3, 6

0 in equation (42), according

to the definitions (18), (19) ^and (20) ^{of those} parameters. The result is

When one coupling Ki

0, ^{that is} ci

1 and si

0, ^{there is no} interaction between planes ^{and we} obtain,

from (43), ^the following expressions ^{of the} partition ^functions

Moreover, ^for equal couplings (ci

C2

C3

c, si = S 2

S3

s) ^{in the} planes ^{we have}

a result to be compared with the solution by Onsager and Kaufman [5] ^{for an} isotropic square lattice, ^that reduces to

for a 2 ^x 2 lattice, using ^{our notation. Hence, the} following ^{relation links the two} and three-dimensional partition

functions

as would be expected.

4. Thermodynamical functions of the 2 x 2 x 2 isotropic Ising system.

Having performed ^this simple check of the general expression (42), ^{we now} proceed ^{to the} isotropic ^{case :} K1 = K2

K3

^{K, and} readily obtain from (42), ^{first the} partition function with field

and second the partition ^{function without field}

In order to be fully ^{aware of} the difficulties induced by ^the presence of a field in three-dimensional Ising systems, ^{it is} certainly instructive to compare equations (42) ^to (43), ^or equations (52) ^to (51).

The specific ^heat per spin is obtained from the expression

where Z8 ^and Zg ^{are the} first and second derivative with respect ^{to c of} expression (52). ^A numerical calculation of C as a function of T shows that the specific ^{heat is a} slowly varying function of T with a maximum. This behaviour is qualitatively analogous ^{to the} specific ^{heat variation of the 2 x 2 lattice} [6]. ^{In the 2 x 2 x 2 case,}

the maximum C/8 kB

^{0.61119 was found for} kB TjJ

^{3.57066 or K}

^{0.28006, that is not so} far from the most accepted ^value [7] ^{for the critical} coupling of the infinite three-dimensional Ising ^{model in} zero-field, K,

^0.22166.

Concerning ^the magnetization, it is obtained by differentiating equation (51) ^with respect ^{to h as}

We note that

dc’ldh

(kB T)-1 s’ ^and ds’/dh

(kB T)-1 ^{c’ and} upon setting C3

^c in equations (18) (20), ^{we have}

Using those relations we find

and note that M(O)

^{0; the} system being finite does not exhibit spontaneous magnetization.

In conclusion, through the full solution of a simple ^{model, we} have shown that exact analytic ^solutions

of finite-size three-dimensional Ising models could benefit from the above definition of a new matrix product

used in connection with three other classical products. It will be shown in a forthcoming paper that larger ^{N x N x}

N lattices can benefit from the present scheme which summarily consists in three steps : ^{first a summation over}

(N ^{x N x} N)/2 spins by using ^the special ^matrix product, ^{second a reduction to} N/2 ^terms by ^{means of the} Hada-

mard product, ^third the determination of the trace of the transfer matrix by taking advantage ^{of the} properties

of the Pauli matrices.

1126

References

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

252.

[2] ^{NOVOTNY, M.} A., ^{J. Math.} Phys. ²⁰ (1979) ^1146.

[3] ^{PEARSON, R. B.,} Phys. ^{Rev. 26} (1982) ^{6285 and}

ITZYKSON, C., PEARSON, ^R. B., ZUBER, ^J. B., ^Nucl.

Phys. ^{B 220} (1983) ^415.

[4] HALMOS, ^P. R., Finite-dimensional vector spaces (Prince- ton, N. J. : Van Nostrand) ^1958.

[5] KAUFMANN, B., Phys. ^{Rev. 76} (1946) ^1232.

[6] FERDINAND, ^A. E., FISHER, M. E., Phys. ^{Rev. 185} (1969)

832.

[7] PAWLEY, ^G. S., SWENDEN, ^R. H., WALLACE, ^D. J.,

WILSON, ^K. G., Phys. ^{Rev. B 29} (1984) ^4030.