Ising's roots and the transfer-matrix eigenvalues

TL;DR

This paper revisits Ernst Ising's original methods for solving the 1D Ising model, revealing that the roots of certain polynomials, derived combinatorially, are equivalent to the transfer matrix eigenvalues, and discusses extensions to multi-state models.

Contribution

It demonstrates the equivalence between Ising's roots and transfer-matrix eigenvalues and explores the generalization to a three-state model, expanding understanding of Ising's original solutions.

Findings

Ising's roots match transfer matrix eigenvalues

Explicit connection between combinatorial and algebraic methods

Extension to three-state models as a precursor to multi-component systems

Abstract

Today, the Ising model is an archetype describing collective ordering processes. And, as such, it is widely known in physics and far beyond. Less known is the fact that the thesis defended by Ernst Ising 100 years ago (in 1924) contained not only the solution of what we call now the `classical 1D Ising model' but also other problems. Some of these problems, as well as the method of their solution, are the subject of this note. In particular, we discuss the combinatorial method Ernst Ising used to calculate the partition function for a chain of elementary magnets. In the thermodynamic limit, this method leads to the result that the partition function is given by the roots of a certain polynomial. We explicitly show that `Ising's roots' that arise within the combinatorial treatment are also recovered by the eigenvalues of the transfer matrix, a concept that was introduced much later. Moreover, we discuss the generalization of the two-state model to a three-state one presented in Ising's thesis, but not included in his famous paper of 1925 (E. Ising, Z.Physik 31 (1925) 253). The latter model can be considered as a forerunner of the now abundant models with many-component order parameters.