Relativistic Landau Models and Generation of Fuzzy Spheres

TL;DR

This paper explores how level projection in relativistic Landau models can generate fuzzy geometries, including fuzzy spheres and super fuzzy geometries, with applications to graphene systems.

Contribution

It introduces a novel application of level projection to relativistic Landau models to generate and analyze fuzzy geometries, including the effects of mass deformation and supersymmetry.

Findings

Derived explicit eigenstates of the Dirac-Landau operator.

Established an $SU(2)$ gauge transformation linking relativistic and non-relativistic models.

Generated fuzzy geometries from relativistic Landau levels and applied to graphene.

Abstract

Non-commutative geometry naturally emerges in low energy physics of Landau models as a consequence of level projection. In this work, we proactively utilize the level projection as an effective tool to generate fuzzy geometry. The level projection is specifically applied to the relativistic Landau models. In the first half of the paper, a detail analysis of the relativistic Landau problems on a sphere is presented, where a concise expression of the Dirac-Landau operator eigenstates is obtained based on algebraic methods. We establish $SU(2)$ "gauge" transformation between the relativistic Landau model and the Pauli-Schr\"odinger non-relativistic quantum mechanics. After the $SU(2)$ transformation, the Dirac operator and the angular momentum operastors are found to satisfy the $SO(3,1)$ algebra. In the second half, the fuzzy geometries generated from the relativistic Landau levels are elucidated, where unique properties of the relativistic fuzzy geometries are clarified. We consider mass deformation of the relativistic Landau models and demonstrate its geometrical effects to fuzzy geometry. Super fuzzy geometry is also constructed from a supersymmetric quantum mechanics as the square of the Dirac-Landau operator. Finally, we apply the level projection method to real graphene system to generate valley fuzzy spheres.