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Journal of Mathematical Physics

Algebraic Derivation of the Partition Function of a Two-Dimensional Ising Model

Colin J. Thompson

Published Sep 1, 1965• 6 citations

Abstract

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.

DOI: 10.1063/1.1704789

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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.

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

{2}^{n}

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

m

-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 ${n}^{\ensuremath{-}2}$.

The computation of classical Ising partition functions, coming from statistical physics, is a natural generalization of binary optimization. This is a notoriously hard problem in general, which makes it an especially interesting task to consider in the search for practical quantum advantage in near term quantum computers. In this work we view classical Ising models (on certain graphs) as quantum imaginary time evolution, which is enabled by the use of the transfer matrix mapping. We study this mapping from two points of view: (1) following Onsager and Kaufman’s original solution of the 2D Ising model, which serves as a starting point, we consider more general models and the possibility of a similar Lie-theoretic solution; (2) we consider quantum algorithms for the computation of partition functions and thermal averages via transfer matrices, which can be implemented either with block encodings inside larger unitaries or by approximating the state trajectories with unitary operators.

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

2^m\times 2^n

(

m,n\in\N

), a variety of recursion relations in

m,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.