Local Thermodynamical Stability of Fermion Lattice Systems

Abstract

Within the framework for equilibrium statistical mechanics of Fermion lattice systems formulated in our preceding work, we study the local thermodynamical stability (LTS) as an alternative characterization of equilibrium states, which works without the translation invariance assumption for the states.

We propose two versions, called LTS-M (mathematical) and LTS-P (physical) according to the choice of the algebra of the outside system for a local region I, LTS-M for the commutant of the local subalgebra A(I) and LTS-P for the subalgebra A(I^c) for the complementary region I^c of I. We show that the two conditions are equivalent for even states, evenness referring to Fermion numbers.

By applying known methods of proof by Sewell and Araki, the following results are obtained: (1) The LTS-M condition implies the dKMS condition for a general state ϕ for an arbitrary general potential (in our technical sense). The same statement holds for the LTS-P condition if ϕ is even. (2) The LTS-M or LTS-P condition for a translation invariant state implies that the state is a solution of the variational principle for any translation covariant standard potential.