Riemannian Geometry of Ising Model in the Bethe Approximation

A method combining statistical equilibrium theory and metric geometry is used to study thermodynamic scalar curvature in the neighborhood of the Curie critical temperature for an Ising model of ferromagnetism. Using a Bethe type free energy expression, a non-diagonal metric is introduced on the two-dimensional phase space of long-range and short-range order parameters. Based on the metric elements Christoffel symbols, curvature tensor and Ricci tensor are found. An expression (containing equilibrium order parameters) is derived for Riemann scalar curvature (R). Its behavior near the critical temperature is examined analytically. We find that R tends toward plus infinity while approaching the critical point. This result fits well with those in the exact one-dimensional chain and mean-field Ising model in the lowest order approximation.