Alternative description of the 2D Blume-Capel model using Grassmann algebra

TL;DR

This paper employs Grassmann algebra to analyze the phase transition in the 2D Blume-Capel model, mapping it onto an effective fermionic theory to identify critical points and compare with numerical simulations.

Contribution

It introduces a novel fermionic representation of the 2D Blume-Capel model, extending the free fermion Ising action with a quartic interaction to study phase transitions.

Findings

Exact mass of the fermionic theory obtained

Critical line defined by vanishing mass condition

Quartic interaction causes fermion spectrum instability at tricritical point

Abstract

We use Grassmann algebra to study the phase transition in the two-dimensional ferromagnetic Blume-Capel model from a fermionic point of view. This model presents a phase diagram with a second order critical line which becomes first order through a tricritical point, and was used to model the phase transition in specific magnetic materials and liquid mixtures of He$^3$-He$^4$. In particular, we are able to map the spin-1 system of the BC model onto an effective fermionic action from which we obtain the exact mass of the theory, the condition of vanishing mass defines the critical line. This effective action is actually an extension of the free fermion Ising action with an additional quartic interaction term. The effect of this term is merely to render the excitation spectrum of the fermions unstable at the tricritical point. The results are compared with recent numerical Monte-Carlo simulations.