Thermal States in Conformal QFT. II

TL;DR

This paper studies the structure of KMS states in conformal quantum field theory, revealing the existence of multiple states in non-rational models and providing classifications for specific nets, extending previous results on uniqueness.

Contribution

It extends the analysis of KMS states to non-rational conformal nets, classifies states on specific models, and introduces a variation of the Araki-Haag-Kastler-Takesaki theorem for locally normal states.

Findings

Unique KMS state for rational nets

Multiple KMS states for non-rational nets

Complete classification for U(1)-current and Virasoro c=1 nets

Abstract

We continue the analysis of the set of locally normal KMS states w.r.t. the translation group for a local conformal net A of von Neumann algebras on the real line. In the first part we have proved the uniqueness of KMS state on every completely rational net. In this second part, we exhibit several (non-rational) conformal nets which admit continuously many primary KMS states. We give a complete classification of the KMS states on the U(1)-current net and on the Virasoro net Vir_1 with the central charge c=1, whilst for the Virasoro net Vir_c with c>1 we exhibit a (possibly incomplete) list of continuously many primary KMS states. To this end, we provide a variation of the Araki-Haag-Kastler-Takesaki theorem within the locally normal system framework: if there is an inclusion of split nets A in B and A is the fixed point of B w.r.t. a compact gauge group, then any locally normal, primary KMS state on A extends to a locally normal, primary state on B, KMS w.r.t. a perturbed translation. Concerning the non-local case, we show that the free Fermi model admits a unique KMS state.