Quantum Superspace and Bloch Electron Systems with Zeeman Effects: *-Bracket Formalism for Super Curtright-Zachos Algebras

TL;DR

The study addresses how Curtright–Zachos (CZ) algebras, as Hom-Lie deformations of Virasoro, can be supersymmetrized and physically realized in Bloch electron systems under Zeeman effects. It develops a quantum-superspace (QSS) framework covariant under $GL_q(1,1)$ and a magnetic-spin-matrix basis (MSB) correspondence, introducing a mixing mechanism between bosonic and Grassmann components to realize Type 3 (enabling $N=2$) super CZ algebras in both continuous MT and discrete TBM settings. The work provides explicit MT and Weyl/cyclic-matrix representations for the super CZ generators, and a unified $\\ast$-bracket formalism with $Z_2$-grading that coherently connects QSS, MSB, and TBM realizations. This establishes a rigorous bridge between quantum geometry and solid-state systems, with potential extensions to other quantum spaces and deformations, and suggests a framework to explore noncommutative geometry in realistic electron systems under strong magnetic fields.

Abstract

We introduce supersymmetric extensions of the Hom-Lie deformation of the Virasoro algebra (super Curtright-Zachos algebra), as realized in the GL(1,1) quantum superspace, for Bloch electron systems under Zeeman effects. By examining the duality inherent in quantum superspace scaling operators, we establish a correspondence between quantum superspace and its physical realization through a novel operator mixing mechanism. For the continuous case, we construct super Curtright-Zachos algebra using magnetic translations and spin matrix bases, demonstrating explicit realizations for both N=1 and N=2 supersymmetric algebras with a natural N=2 decomposition. For the discrete case, we establish cyclic matrix representations in tight-binding models. We organize these structures through the *-bracket formalism with Z2-grading, revealing how the quantum superspace structure manifests in physical systems while preserving essential algebraic properties.