  A model in statistical mechanics, characterised by the corresponding Gibbs
measure, is a subset of the totality of probability distributions on the phase
space. The shape of this subset, i.e., the geometry, then plays an important
role in statistical analysis of the model. It is known that this subset has the
structure of a manifold equipped with a Riemannian metric, given by the Fisher
information matrix. Invariant quantities such as thermodynamic curvature have
been studied extensively in the literature, although a satisfactory physical
interpretation of the geometry has not hitherto been established. In this
article, we investigate the thermodynamic curvature for one and two dimensional
Ising models and report the existence of a geometric phase transition
associated with a change in the signature of the curvature. This transition is
of a continuous type, and exists for finite systems. The effect may be tested
in principle by mesoscopic scale experiments.
