Linear Fermion Systems, Molecular Field Models, and the KMS Condition

We take advantage of the simplifications made possible by assuming the system to be infinite from the beginning, to study some exactly solvable models in statistical mechanics. (1) For fermi lattice systems with quadratic hamiltonians the unifying concept of our treatment is a precise formulation of the idea of a linear system. This concept makes it possible to define a dynamical matrix which incorporates all the information one needs for equilibrium and non‐equilibrium problems. As an example, we consider the alternating X Y chain. Furthermore we pay some attention to ergodicity, pointing out the similarity to the infinite harmonic crystal. (2) For molecular field models, including the BCS model, the method is to consider an arbitrary extremal homogeneous state, to construct the dynamics in the representation determined by that state, and then to use the KMS condition to single out the equilibrium state of the infinite system. A synthesis of these methods – (1) and (2) – enables us to give a new simple treatment of the BCS model and a direct analysis of its mathematical structure. In addition we show the equivalence of Kac and van der Waals (spin) systems in the so‐called van der Waals limit. Finally we indicate briefly how to deal with molecular field models with polynomial and more general interactions.