|Examples of a Grassmann technique applied to classical spin systems in 2D|
|Friday 15 May 2009 , 09h00|
|Talk at the workshop (2009)|
In this talk, we will present an analytical method using Grassmann techniques to study the fermionic theory underlying general spin-S Ising models in two dimensions. The simplest case S=1/2 was understood a long time ago and is known to be equivalent to a massive free fermion theory possessing a second-order transition when the mass vanishes. Here we try to extend the method for general spin-S with the additional presence of a splitting field in the Hamiltonian. One of our motivation comes from the fact that there is little knowledge about the fermionic theory of such models and the relation to the free fermion theory of the S=1/2 2D Ising model. In particular we can show that the Blume-Capel model can be exactly mapped onto an interacting fermion model, with a bare mass depending on the value of the splitting field. The location of the points that make this bare mass vanish is very close to the numerical results found by Monte-Carlo method in the region of the second-order transition. The extension to spin-S models and a method to find the bare mass in general is presented, with an accuracy of less than 1% for the second-order transition lines in these models. The particular values of the transition points at zero splitting field are then compared to high and low-temperature expansions.