On the limit Gibbs states of the spherical model

Abstract

The effect of a class of Hamiltonian perturbations, vanishing in the thermodynamic limit, on the limit Gibbs states of the spherical and mean spherical models is studied. The perturbation term is taken in the form of interaction energy with uniform magnetic field of strength h0N- alpha , where h0 in r1 and alpha )0 are parameters, and N is the number of particles. For fixed temperatures below the critical temperature, in the absence of constant external magnetic fields and at alpha =1 the authors obtain convex sets of different mixed Gibbs states parametrised by h0. A natural one-parameter generalisation of the Kac-Thompson transformation kernel which relates the states of the mean spherical model to the states of the spherical model is found. When 0( alpha (1 and h0 not=0, or alpha =1 and h0=+or- infinity , this kernel becomes a delta function even below the critical temperature; then the states in both ensembles coincide with each other and with one of the two (depending on the sign of h0) extreme points. The case of alpha >1 is found to lead to the well known results corresponding to the absence of perturbation (h0=0).