Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant D_PQ

TL;DR

This paper derives an exact expression for the spontaneous magnetization of the superintegrable chiral Potts model by representing the relevant determinant as a product of Cauchy-like matrices, confirming a conjecture from 1989.

Contribution

It introduces a novel method to evaluate the determinant D_PQ using Cauchy-like matrices, providing a new exact calculation approach for the model's magnetization.

Findings

Derived an exact formula for D_PQ as a product of Cauchy-like matrices

Confirmed the 1989 conjecture on spontaneous magnetization

Provided a new computational method for similar determinants

Abstract

For the Ising model, the calculation of the spontaneous magnetization leads to the problem of evaluating a determinant. Yang did this by calculating the eigenvalues in the large-lattice limit. Montroll, Potts and Ward expressed it as a Toeplitz determinant and used Szego's theorem: this is almost certainly the route originally travelled by Onsager. For the corresponding problem in the superintegrable chiral Potts model, neither approach appears to work: here we show that the determinant D_PQ can be expressed as that of a product of two Cauchy-like matrices. One can then use the elementary exact formula for the Cauchy determinant. One of course regains the known result, originally conjectured in 1989.