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Physical Review

Crystal Statistics of a Two-Dimensional Ising Lattice

Gordon Frank Newell

Published May 15, 1950• 11 citations
Publisher

Abstract

The partition function $Z$ of a two-dimensional Ising lattice is evaluated by expressing it in the form $Z=\ensuremath{\Sigma}{i}^{}\ensuremath{\lambda}_{i}^{}{}_{}{}^{\mathrm{nm}}$ where ${\ensuremath{\lambda}}_{i}$ are the eigenvalues of a ${2}^{n}$-dimensional matrix M, $n$ is the number of rows and $m$ the number of columns of the lattice. M is a generalization of the V-shaped matrix studied by Kramers and Wannier in connection with the same problem. It is shown that M can be expressed in terms of ${2}^{n}$-dimensional representations of $2n$-dimensional orthogonal matrices. The eigenvalues of M are determined for $n$ large by making use of the known relations between the eigenvalues of a $2n$-dimensional orthogonal matrix and the eigenvalues of its ${2}^{n}$-dimensional representative.The lattice considered differs only slightly from that treated by Onsager and Kaufman. In the lattice considered here, the boundary effects are eliminated by introducing an interaction between the last lattice point of each row and the first point of the next row. The matrix V considered by Onsager and Kaufman is, except for the boundary effects, equivalent to the matrix ${\mathbf{M}}^{n}$. $Z$ is shown to differ from the partition function calculated by Onsager and Kaufman only in order ${n}^{\ensuremath{-}2}$.

DOI
10.1103/physrev.78.444

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World Scientific series in 20th century physics•Nov 1, 1995•3 citations
The partition function ZZZ of a two-dimensional Ising lattice is evaluated by expressing it in the form Z=\ensuremathΣi\ensuremathλinmZ=\ensuremath{\Sigma}{i}^{}\ensuremath{\lambda}_{i}^{}{}_{}{}^{\mathrm{nm}}Z=\ensuremathΣi\ensuremathλi​​nm where \ensuremathλi{\ensuremath{\lambda}}_{i}\ensuremathλi​ are the eigenvalues of a 2n{2}^{n}2n-dimensional matrix M, nnn is the number of rows and mmm the number of columns of the lattice. M is a generalization of the V-shaped matrix studied by Kramers and Wannier in connection with the same problem. It is shown that M can be expressed in terms of 2n{2}^{n}2n-dimensional representations of 2n2n2n-dimensional orthogonal matrices. The eigenvalues of M are determined for nnn large by making use of the known relations between the eigenvalues of a 2n2n2n-dimensional orthogonal matrix and the eigenvalues of its 2n{2}^{n}2n-dimensional representative.The lattice considered differs only slightly from that treated by Onsager and Kaufman. In the lattice considered here, the boundary effects are eliminated by introducing an interaction between the last lattice point of each row and the first point of the next row. The matrix V considered by Onsager and Kaufman is, except for the boundary effects, equivalent to the matrix Mn{\mathbf{M}}^{n}Mn. ZZZ is shown to differ from the partition function calculated by Onsager and Kaufman only in order n\ensuremath−2{n}^{\ensuremath{-}2}n\ensuremath−2.
In the algebraic formulation of the Ising model, the partition function is expressed as the trace of a power VM of the transfer operator V, or equivalently, as the sum of Mth powers of the eigenvalues of V. In the derivations of Kaufman, Onsager, and more recently of Schultz, Mattis, and Lieb (SML), the transfer operator is first reduced to a more amenable form for computation and then, in principle at least, diagonalized. For the infinite lattice, only the largest eigenvalue of V is needed, and this is all Onsager and SML compute. Kaufman finds all the eigenvalues and is thus able to write down the partition function for the finite lattice. In the present work we give an alternative derivation of the SML form for V, and show how the Kaufman result can be obtained from this form without actual diagonalization. Instead of diagonalizing V, the evaluation of the trace is done directly after assigning a simple representation to V.
An approximation for calculating the thermodynamic properties of spin-\textonehalf{} Ising models is described. The method is based on the formulation of Kramers and Wannier for the square lattice, and involves expressing the free energy in terms of the limiting maximum eigenvalue of a matrix sequence which represents the contribution from a single spin of the lattice. The application of this formulation to various two- and three-dimensional lattices shows that many of these are characterized by an identically structured matrix sequence with exactly two nonvanishing elements in each row and column. To explore the potential of using this similarity and its simplicity for analyzing Ising models we consider the two-dimensional square lattice in an external field and approximate the doubly infinite lattice by a semi-infinite one consisting of $n (\ensuremath{\le}10)$ infinitely long rows. Calculations are carried out and compared with (1) Onsager's formula for the free energy, (2) the Onsanger-Yang formula for the spontaneous magnetization, and (3) values calculated from series. Except for the region around the critical point where the correlation length exceeds nnn lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.
The partition function of the two-dimensional Ising model on a square lattice with nearest-neighbour interactions and periodic boundary conditions is investigated. Kaufman [Phys. Rev. 76, 1232--1243 (1949)] gave a solution for this function consisting of four summands. The summands are rewritten as functions of a low-temperature expansion variable, resulting in polynomials with integer coefficients. Considering these polynomials for system sizes 2m×2n2^m\times 2^n2m×2n (m,n∈Nm,n\in\Nm,n∈N), a variety of recursion relations in m,nm,nm,n are found. The recursions reveal a rich structure of the partition function and can be employed to render the computer algebra calculation of the microcanonical partition function more efficient.
The low-temperature series expansion for the partition function of the two-dimensional Ising model on a square lattice can be determined exactly for finite lattices using Kaufman's generalization of Onsager's solution. The exact distribution function for the energy can then be determined from the coefficients of the partition function. This provides an exact solution with which one can compare energy histograms determined in Monte Carlo simulations. This solution should prove useful for detailed studies of statistical and systematic errors in histogram reweighting.
The derivation of thermodynamic and magnetic properties of a crystal, on the basis of the Ising model, concerns essentially two non-commuting matrices which are here called A and B. In particular, one must evaluate the traces of Am, Bm, AmBm, and (AB)m for general integer m. The first two traces relate to the one-dimensional case; the third and fourth to the two-dimensional case. A and B are of dimension 2n, m and n being respectively the numbers of rows and columns.Previous methods of evaluating these traces have been elaborate, involving concepts such as spinor rotations in a space of 2n dimensions. In this paper elementary methods are used to derive closed expressions for the first three and the fourth is evaluated in an interesting special case.To date the Ising model has been treated exactly in two cases only. For this reason the search for simple calculational methods is of particular importance.
mathematical theory of order-disorder transition. The problem of determining the partition function for a lattice with simple interaction model was found, by severai authors,!) to be reduced to an eigenvalue problem which is easier to handle with,and which, with suitable approximations, would reveal more refined details of the phase transition than the former theories. This method of attack culminated in the work of Onsager,2) who succeeded In giving the rigorous solution of the above mentioned eigenvalue problem for two-dimensional simple square Ising lattice. The mathematical tools involved, however, are so hopelessly complicated that one would quite simply lose sight in the jungle of hypercomplex numbers. A con­ siderable improvement on this mathematical point has been attainp.d by Husimi and Syozi3) who used it to solve the eigenvalue problem for the honeycomb type lattice. The present author, independently of them,' has reached more or less similar ideas and considerations. which it is the purpos~ of the present paper to expose in brief detail. Though as yet no substantial applications has been attempted, nor anything physically new has been derived, it may be hoped that it will do some profit for those who are interested in such problems.