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, ℛ, of this metric can be calculated explicitly in the thermodynamic limit and is found to be ℛ = 1 + cosh( h )/√sinh 2 ( 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 ℛ for the one-dimensional q -state Potts model finding an expression of the form ℛ = A ( q , β, h ) + B ( q , β, h )/√η( q , β, h ), where η( q , β, h ) is the Potts analogue of sinh 2 ( 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.