Deformed CCR/CAR and Free Monotone Transport: Quons and Fock Parafermions

Abstract

This thesis consists of two parts, both centred around the theme of deformations of the canonical (anti-)commutation relations. In the first part, we provide an accessible and self-contained, yet complete, account of the recent partial resolution of the isomorphism problem concerning the q-Gaussian von Neumann algebras of Bozejko and Speicher by the free monotone transport method of Guionnet and Shlyakhtenko. We cover the necessary background in free probability, describe the construction of q-Gaussians and prove some of their elementary properties due to Ricard and Voiculescu, and provide a detailed proof of the existence of free monotone transport and its application to q-Gaussians. Some special attention is given to the commutative case and its links to random matrices, large deviations, and optimal transportation. The second part starts with a brief review of the physics of quons, which are the particles in the Fock space realisation of the q-Gaussians. Afterwards, we turn to parafermions and their importance to edge modes and topological phases. The Fock parafermion operators due to Cobanera and Ortiz are introduced, which allow for a Fock representation of parafermions. Then, we study Fock parafermions as fundamental degrees of freedom, considering both Potts-like Hamiltonians in the spirit of Calzona et alia, and simple tight-binding Hamiltonians in the spirit of Rossini et alia. This includes mappings of Fock parafermions to electrons and mixed Fermion-Boson systems, the exact ground states for the Potts-like models, analytical evidence that the tight-binding models generally have conformal charge c = 1, and first steps towards constructing their phase diagram.