The information geometry of the one-dimensional Potts model

In various statistical-mechanical models the introduction of a metric into the space of parameters (e.g. the temperature variable, β, and the external field variable, h, in the case of spin models) gives an alternative perspective on the phase structure. For the one-dimensional Ising model the scalar curvature, Script R, of this metric can be calculated explicitly in the thermodynamic limit and is found to be Script R = 1 + cosh(h)/√sinh2(h) + exp(−4β). This is positive definite and, for physical fields and temperatures, diverges only at the zero-temperature, zero-field 'critical point' of the model. In this paper we calculate Script R for the one-dimensional q-state Potts model finding an expression of the form Script R = A(q, β, h) + B(q, β, h)/√η(q, β, h), where η(q, β, h) is the Potts analogue of sinh2(h) + exp(−4β). This is no longer positive definite, but once again it diverges only at the critical point in the space of real parameters. We remark, however, that a naive analytic continuation to complex field reveals a further divergence in the Ising and Potts curvatures at the Lee–Yang edge.