Proof of the determinantal form of the spontaneous magnetization of the superintegrable chiral Potts model

TL;DR

This paper proves that the spontaneous magnetization of the superintegrable chiral Potts model can be expressed as a determinant, confirming a conjecture and linking algebraic properties to known results.

Contribution

The paper provides a rigorous proof that the sum over matrix elements for the magnetization equals a determinant, establishing a key algebraic property of the model.

Findings

Confirmed the determinant form of the spontaneous magnetization

Validated the conjecture relating sum and determinant expressions

Connected algebraic structures to known physical results

Abstract

The superintegrable chiral Potts model has many resemblances to the Ising model, so it is natural to look for algebraic properties similar to those found for the Ising model by Onsager, Kaufman and Yang. The spontaneous magnetization M_r can be written in terms of a sum over the elements of a matrix S_r. The author conjectured the form of the elements, and this conjecture has been verified by Iorgov et al. The author also conjectured in 2008 that this sum could be expressed as a determinant, and has recently evaluated the determinant to obtain the known result for M_r. Here we prove that the sum and the determinant are indeed identical expressions.