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

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.

32 pages, 2 figures;

References in corpus (12)

• The Critical Behaviour of the Spin-3/2 Blume-Capel Model in Two Dimensions
• Tricritical universality in a two-dimensional spin fluid
• Universality and scaling study of the critical behavior of the two-dimensional Blume-Capel model in short-time dynamics
• Exact partition functions of the Ising model on MxN planar lattices with periodic-aperiodic boundary conditions
• Universal Amplitude Ratios of The Renormalization Group: Two-Dimensional Tricritical Ising Model
• Charge orderings in the atomic limit of the extended Hubbard model
• Fermionic Structure of Two-Dimensional Ising Model with Quenched Site Dilution
• Universal Ratios in the 2-D Tricritical Ising Model
• Bosonization and even Grassmann variables
• Anticommuting Integrals and Fermionic Field Theories for Two-Dimensional Ising Models
• 1D action and partition function for the 2D Ising model with boundary magnetic field
• Boundary field induced first order transition in the 2D Ising model: Exact study

Cited by in corpus (8)

• Multicriticality in the Blume-Capel model under a continuous-field probability distribution
• The random field Blume-Capel model revisited
• Exact solution of the $2d$ dimer model: Corner free energy, correlation functions and combinatorics
• Spin-liquid behaviour and the interplay between Pokrovsky-Talapov and Ising criticality in the distorted, triangular-lattice, dipolar Ising antiferromagnet
• Fermions and Disorder in Ising and Related Models in Two Dimensions
• Second-order critical lines of spin-S Ising models in a splitting field with Grassmann techniques
• Short-time dynamics in phase-ordering kinetics
• Information-Geometric Signatures from Nonextensivity in the $1$-D Blume-Capel Model