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

Abstract

We discuss the Curtright-Zachos (CZ) deformation of the Virasoro algebra and its extensions 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 ⁎-bracket commutator. The phase factors can be formulated as a ⁎-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.