Algebraic reduction of the Ising model

TL;DR

This paper presents an algebraic method to reduce the calculation of the Ising model's spontaneous magnetization on a cylindrical lattice, simplifying determinants and enabling comparisons with related models.

Contribution

It introduces a Clifford algebra-based reduction technique that transforms large determinants into smaller ones, facilitating analysis of the Ising model and related models.

Findings

Partition functions expressed as L by L determinants

Reduction to approximately L/2 by L/2 determinants

Potential for algebraic calculation of the superintegrable chiral Potts model's magnetization

Abstract

We consider the Ising model on a cylindrical lattice of L columns, with fixed-spin boundary conditions on the top and bottom rows. The spontaneous magnetization can be written in terms of partition functions on this lattice. We show how we can use the Clifford algebra of Kaufman to write these partition functions in terms of L by L determinants, and then further reduce them to m by m determinants, where m is approximately L/2. In this form the results can be compared with those of the Ising case of the superintegrable chiral Potts model. They point to a way of calculating the spontaneous magnetization of that more general model algebraically.