Geometry and Thermodynamic Fluctuations of the Ising Model on a Bethe Lattice

Abstract

A metric is introduced on the two dimensional space of parameters describing the Ising model on a Bethe lattice of co-ordination number q. The geometry associated with this metric is analysed and it is shown that the Gaussian curvature diverges at the critical point. For the special case q=2 the curvature reduces to an already known result for the one dimensional Ising model. The Gaussian curvature is also calculated for a general ferro-magnet near its critical point, generalising a previous result for t>0. The general expression near a critical point is compared with the specific case of the Bethe lattice and a subtlety, associated with the fact that the specific heat exponent for the Bethe lattice vanishes, is resolved.