Abstract
We compare the dynamical characterization of pure thermodynamical phases as extremal KMS states and their characterization as extremal time- or space-invariant states. We find that, for a class of Weiss-Ising models with periodic potentials, the extremal KMS states coincide exactly with the solutions of the self-consistency equations familiar from molecular field methods. We show that the models considered are not η-asymptotically Abelian in time. We conclude that the characterization of pure thermodynamical phases as extremal KMS states is the only correct one for these models. We pay special attention (in particular, in the decomposition of an arbitrary KMS state into its extremal KMS components) to the fact that the time evolution is not an automorphism of the C*-algebra of the quasilocal observables.
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