BCS-spin-model, its thermodynamic representations and automorphisms

Abstract

In this paper two equivalent explicit realizations of the thermodynamic representations of the BCS-spin-model are constructed and since they are of well known form their type is easily determined. Time evolution and vacuum properties of these representations are studied and it is shown that in general the time development of the system is representation-dependent.

Classical Systems in Quantum Mechanics

“…The equilibrium states of the considered system can be defined as the thermodynamic limits of the (unique) Gibbs states of local systems (A J ; τ J ), |J| < ∞, where τ J t ∈ * -Aut A J is generated by the local Hamiltonians Q J defined in (6.1.1). According to [167], these thermodynamic limits coincide with ω n T for T ≥ T c , whereas for 0 < T < T c the limit J → Π leads to the state ω s T .…”

“…model of superconductivity 6.5.1. We shall illustrate in this section the above developed theory by description and analysis of a perhaps simplest nontrivial and physically interesting mathematical model: The strong coupling version of the Bardeen-Cooper-Schrieffer model of the phenomenon of superconductivity in the quasi spin formulation; it was formulated and analyzed in [311,310,167], in the framework of the traditional QM formalism. It can be presented, completed, and solved in the framework of the constructions of the present work as follows:…”

“…Let ω n 0 ∈ K ∞ ⊂ S(A) corresponds to the value F ω from (6.5.31), and let ω s 0 be given by (6.5.26) with T = 0 and with ω F 0 ∈ EK ∞ corresponding to any value of F given in (6.5.32). According to [167], the thermodynamic limit of the (unique) local ground states on A J corresponding to the Hamiltonians Q J coincides with ω s 0 .…”

Non-translation Invariant Product States of Classical Lattice Systems

“…model [9], •••) for which rigorous proofs of the validity of the gap equation method of computing the equilibrium state don't exist yet [10,11].…”