Characterisations of phase transitions in Ising spin systems

Abstract

We construct aC*-algebraic formalism designed to provide a framework for the characterisation of phase transitions in a class of Ising spin systems: this class is large enough to include the rectangular lattice models, of arbitrary finite dimensionality, with nearest neighbour interactions. Using an extension of Onsager's transfer matrix formalism, we express properties of a Gibbs state of a system in terms of a contractive linear transformation, υ₀, of a certain Hilbert space, the properties of υ₀ being governed by the temperature as well as the interactions in the system. We obtain conditions on υ₀ under which the system exhibits a phase transition characterised by (A) a thermodynamical singularity, (B) a change in symmetry, associated with theG-ergodic decomposition of Gibbs states, (C) a divergence of a “correlation length” (appropriately defined) at the critical point, and (D) “scaling laws” in the critical region. Applying our formalism to the rectangular two-dimensional Ising model with nearest neighbour interactions, we show that its phase transition possesses the properties (B) and (C), as well as (A).