Series studies of the Potts model. I: The simple cubic Ising model

TL;DR

This paper extends the finite lattice series expansion method to the three-dimensional Potts model on a simple cubic lattice, providing new series data and critical exponent analysis for the Ising case.

Contribution

It generalizes the series expansion technique to 3D Potts models and extends existing series for the Ising case, enabling better critical behavior analysis.

Findings

Exponential growth of computational effort with series terms in 3D

Extended low-temperature series for Ising model to order 26

Extended high-temperature series for Ising model to order 22

Abstract

The finite lattice method of series expansion is generalised to the $q$-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising ($q=2$) case we have extended low-temperature series for the partition functions, magnetisation and zero-field susceptibility to $u^{26}$ from $u^{20}$. The high-temperature series for the zero-field partition function is extended from $v^{18}$ to $v^{22}$. Subsequent analysis gives critical exponents in agreement with those from field theory.