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

Abstract

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.

Charges and Symmetries in Quantum Theories without Locality

“…This was explained by M O R C~O and STROCCHI [13], who also used their formalism t o compute the plasmon energy in the jellium model [35]. Rather than t'rying to provide an exposition of the theory, we shall attempt to explain it through a particular example, the strong-coupling BCS wodel [45], following a slightly different approach [20]. Although the R.C.S.…”

“…HBo is a self-adjoint operator only in the representation17,. Werefer to [20] for a general way of obtaining these results, but the basic procedure is the following.…”

Quantum Stochastic Processes

“…They define a limiting process; however we expect this limiting process to be neither stationary nor Markov in general, but rather to exhibit the structure with time-dependent Hamiltonians described in the notes to Section 2.4 (cf. van Hemmen [54] Appendix).…”

A new method for the thermomodynamics of the BCS model

“…The BCS model has many simplifying features when studied in the thermodynamic limit from the start (see for example [16,17]). However this approach is not in the same spirit as in this paper since the methods used are algebraic, although in some cases not explicitly so.…”