Symmetry breaking on a finite euclidean lattice

Abstract

In this paper we introduce a new way to implement one-loop euclidean perturbation theory on finite periodic lattices. It is shown that this technique gives both 〈 ϕ〉=0 and 〈 ϕ〉 ≠ 0 solutions for the SO( N )-invariant λφ 4 models, despite the fact that there can be no phase transition on such lattices. In obtaining these results we introduce a self-consistency condition which plays a crucial role on finite lattices. We compare the perturbative results obtained in this way with data obtained from stochastic simulations for the same models, in which we use symmetry transformations to eliminate the drifting of the direction of alignment of the fields. We find that some of the perturbative results agree well with the numerical with the numerical data even for values of the parameters for which one would not expect perturbation theory to work. In the region which becomes the broken-symmetric phase in the continuum limit, the numerical data agrees very well with the 〈 ϕ〉 ≠ 0 solution. This solution does not exist, however, in the whole parameter space. The region where it does not exist becomes the symmetric phase in the continuum limit, and there we find that it is the 〈 ϕ〉=0 solution that fits very well some of, but not all, the numerical data.