Equilibrium Statistical Mechanics of Fermion Lattice Systems
  We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the non-commutativity of local algebras for disjoint regions.   Our major result is the equivalence of the KMS condition and the variational principle with a minimal assumption for the dynamics and without any explicit assumption on the potential. It holds also for spin lattice systems as well, yielding a vast improvement over known results.   All formulations are in terms of a C*-dynamical systems for the Fermion (CAR) algebra with all or a part of the following assumptions:   (I) The interaction is even with respect to the Fermion number. (Automatically satisfied when (IV) below is assumed.)   (II) All strictly local elements of the algebra have the first time derivative.   (III) The time derivatives in (II) determine the dynamics.   (IV) The interaction is lattice translation invariant.   A major technical tool is the conditional expectation from the total algebra onto the local subalgebra for any finite subset of the lattice, which induces a system of commuting squares. This technique overcomes the lack of tensor product structures for Fermion systems and even simplifies many known arguments for spin lattice systems. 
Huzihiro Araki and Hajime Moriya