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

Maxime Clusel (1) , Jean-Yves Fortin (2) , Vladimir Plechko (3)

1 CSMR - Center for Soft Matter Research [New-York]
2 LPTH - Laboratoire de Physique Théorique
3 BLTP - Bogoliubov Laboratory of Theoretical Physics [Dubna]

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. 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.

Domaines: Mécanique statistique [cond-mat.stat-mech]

Maxime Clusel, Jean-Yves Fortin, Vladimir Plechko. Alternative description of the 2D Blume-Capel model using Grassmann algebra. Journal of Physics A: Mathematical and Theoretical, 2008, 41 (40), pp.405004. ⟨10.1088/1751-8113/41/40/405004⟩. ⟨hal-00641685⟩