Hom-Lie-Virasoro symmetries in Bloch electron systems and quantum plane in tight binding models

TL;DR

This paper explores the deformation of the Virasoro algebra using magnetic translation groups in Bloch electron systems, revealing connections to quantum planes and describing certain Hamiltonians with these deformed symmetries.

Contribution

It introduces a novel application of the Curtright-Zachos deformation in the context of Bloch electron systems and links it to quantum plane structures and magnetic translation operators.

Findings

CZ generators are composed of magnetic translation operators.

Phase factors relate to quantum plane deformation parameter q.

Certain TBM Hamiltonians are described by CZ generators.

Abstract

We discuss the Curtright-Zachos (CZ) deformation of the Virasoro algebra and its extentions in terms of magnetic translation (MT) group in a discrete Bloch electron system, so-called the tight binding model (TBM), as well as in its continuous system. We verify that the CZ generators are essentially composed of a specific combination of MT operators representing deformed and undeformed $U(1)$ translational groups, which determine phase factors for a $\ast$-bracket commutator. The phase factors can be formulated as a $\ast$-ordered product of the commutable $U(1)$ operators by interpreting the AB phase factor of discrete MT action as fluctuation parameter $q$ of a quantum plane. We also show that some sequences of TBM Hamiltonians are described by the CZ generators.