Second-order critical lines of spin-S Ising models in a splitting field with Grassmann techniques

TL;DR

This paper introduces a Grassmann variable-based method to analytically determine second-order critical lines in two-dimensional classical spin-$S$ Ising models under a splitting field, aligning well with numerical results.

Contribution

The authors develop an exact Grassmann technique to derive critical lines for spin-$S$ Ising models, extending previous models and enabling analytical study of phase transitions.

Findings

Critical lines match previous numerical estimates

Method applies to various classical Hamiltonians

Provides exact expression for the bare mass in the field theory

Abstract

We propose a method to study the second-order critical lines of classical spin-$S$ Ising models on two-dimensional lattices in a crystal or splitting field, using an exact expression for the bare mass of the underlying field theory. Introducing a set of anticommuting variables to represent the partition function, we derive an exact and compact expression for the bare mass of the model including all local multi-fermions interactions. By extension of the Ising and Blume-Capel models, we extract the free energy singularities in the low momentum limit corresponding to a vanishing bare mass. The loci of these singularities define the critical lines depending on the spin S, in good agreement with previous numerical estimations. This scheme appears to be general enough to be applied in a variety of classical Hamiltonians.