More on Algebraic Structure of the Complete Partition Function for the $ Z_n $ - Potts Model, Part 1

TL;DR

This paper explores the algebraic structure of the complete partition function for the $Z_n$-Potts model, deriving formulas for traces of generalized Clifford algebra generators to facilitate calculations.

Contribution

It introduces a method to compute partition functions of vector Potts models using traces of generalized Clifford algebra generators, extending the approach beyond the $Z_2$ case.

Findings

Derived formulas for traces of generalized Clifford algebra generators.

Established a computational framework for partition functions in $Z_n$ models.

Discussed algebraic properties controlling the calculation complexity.

Abstract

In this first part of a larger review undertaking the results of the first author and a part of the second author doctor dissertation are presented. Next we plan to give a survey of a nowadays situation in the area of investigation. Here we report on what follows. Calculation of the partition function for any vector potts model is at first reduced to the calculation of traces of products of the generalized clifford algebra generators. The formula for such traces is derived. This enables one, in principle, to use an explicit calculation algorithm for partition functions also in other models for which the transfer matrix is an element from generalized clifford algebra. The method - simple for $Z_2$ case - becomes complicated for $Z_n$, $n>2$, however everything is controlled, in principle, due to knowledge of the corresponding algebra properties and those of generalized cosh function. The discussion of the content of the in statu nascendi second part is to be found at the end of this presentation. This constitutes the last section.