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Studies in Applied Mathematics

On the Transfer Matrix for the Two‐Dimensional Ising Model

Douglas B. Abraham

Published Mar 1, 1971• 56 citations
Publisher

Abstract

The eigenvalue problem for the transfer matrix of the two‐dimensional Ising lattice, with free boundaries perpendicular to the transfer direction, is solved. The mathematical mechanism which produces the asymptotic degeneracy of the eigenvalue spectrum below the bulk transition temperature is different from the case with cyclic boundary conditions. The results are used to calculate the two‐spin correlation function between two spins in the same edge of the lattice, and between two spins in opposite edges.

DOI
10.1002/sapm197150171

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The eigenvalue problem for the transfer matrix of the two‐dimensional Ising lattice, with free boundaries perpendicular to the transfer direction, is solved. The mathematical mechanism which produces the asymptotic degeneracy of the eigenvalue spectrum below the bulk transition temperature is different from the case with cyclic boundary conditions. The results are used to calculate the two‐spin correlation function between two spins in the same edge of the lattice, and between two spins in opposite edges.
Abstract In this chapter we study the solution of the two-dimensional Ising model by means of the transfer matrix. Unlike the methods discussed in the previous chapter, the transfer matrix approach has greater generality and can used to solve exactly other two-dimensional models. Even if the general ideas behind this approach have been explained in Chapter 2 by means of the one-dimensional case, their application to the two-dimensional cases requires more powerful and sophisticated mathematical tools: for instance, the study the eigenvalues of the transfer matrix in the Ising model for T = Tcneeds to employ elliptic functions. The same is also true for other models. In order to present in the simplest possible way the main lines of this method, in the following we focus attention only on the solution of the model at T = Tcbecause this case can be analyzed in terms of simple trigonometric functions.
Some years ago Ashkin and Lamb noted that the phase transition in the two-dimensional Ising model with nearest-neighbor interaction was characterized mathematically by an asymptotic degeneracy of the largest eigenvalue of a linear operator. More recently Kac and Thompson showed this eigenvalue degeneracy also characterized the phase transition in the Kac model (with weak, long-range forces), suggesting that this ``mathematical mechanism'' is not restricted to systems with short-range forces. However both the Kac model and the Ising model consider the ``spins'' to be one-dimensional unit vectors assuming only the discrete values +1 and −1. We are therefore led to consider the nature of the phase transition in one of the few exactly soluble models in which the spins can assume an entire continuum of orientations-the closed linear chain of classical spins of arbitrary spin dimensionality interacting isotropically through the Hamiltonian Hν=− ∑ i=1N Ji,i+1Si(ν)·Si+1(ν),where the exchange constants Ji,i+1 are arbitrary numbers. We find for all spin dimensionalities ν that the two-spin correlation function may be expressed as ρN(ν) (γ) = (λ1(ν)/λ0(ν))r, where λ0(ν) and λ1(ν) are the largest and next-largest eigenvalues of a certain linear operator. Thus the onset of longrange order, limr→∞ρN(ν) (γ)≠0, is characterized by the degeneracy of λ0(ν) and λ1(ν). This onset of long-range order occurs at a critical temperature Tc(ν)=0, for all values of spin dimensionality ν.