Modular Structures on Trace Class Operators and Applications to Themodynamical Equilibrium States of Infinitely Degenerate Systems

TL;DR

This paper investigates the structure of thermal equilibrium states (KMS states) for infinitely degenerate Hamiltonians, focusing on Landau levels, and explores extensions of the KMS concept to non-standard states.

Contribution

It classifies KMS states for infinitely degenerate systems, shows the absence of cyclic and separating vectors for Landau Hamiltonians, and discusses extensions of KMS states beyond normal and semifinite states.

Findings

Classified all KMS states for the Landau Hamiltonian.

Proved the non-existence of cyclic and separating vectors in this context.

Explored extensions of KMS states to non-$\sigma$-additive and singular states.

Abstract

We study the thermal equilibrium states (KMS states) of infinitely degenerate Hamiltonians, in particular, we study the example of the Landau levels. We classify all KMS states in an example of algebra suitable for describing infinitely degenerate systems and we show that there is no cyclic and separating vector corresponding to the Landau Hamiltonian. Then, we try to reproduce the thermodynamical limit of a finite box as used in the very beginning of the theory of KMS states by Haag Hugenholtz and Winnink. Finally, we discuss the situation from the point of view of non-$\sigma$-additive probabilities, non-normal nor semifinite states, singular (Dixmier) states and, hence, an extension of the concept of KMS state.