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

TL;DR

This paper introduces a geometric framework for analyzing the Ising model on a Bethe lattice, revealing how the curvature of parameter space diverges at critical points and generalizing previous results for ferromagnets.

Contribution

It develops a metric on the parameter space of the Ising model on Bethe lattices, analyzes its geometry, and extends curvature calculations to general ferromagnets near criticality.

Findings

Gaussian curvature diverges at the critical point

Curvature reduces to known results for q=2 case

Generalized curvature calculations for ferromagnets near criticality

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.