Quantum mechanics on noncommutative Riemann surfaces

TL;DR

This paper explores the quantum mechanics of a charged particle on a noncommutative Riemann surface with a magnetic field, revealing unique spectral properties depending on the magnetic field strength.

Contribution

It formulates quantum mechanics on noncommutative Riemann surfaces via noncommutative AdS_2 and introduces a quantization condition for single-valued states, extending prior models.

Findings

Discrete Landau levels and continuum in subcritical magnetic fields

Pure Landau level phase in overcritical magnetic fields

No magnetic field quantization on the covering space

Abstract

We study the quantum mechanics of a charged particle on a constant curvature noncommutative Riemann surface in the presence of a constant magnetic field. We formulate the problem by considering quantum mechanics on the noncommutative AdS_2 covering space and gauging a discrete symmetry group which defines a genus-g surface. Although there is no magnetic field quantization on the covering space, a quantization condition is required in order to have single-valued states on the Riemann surface. For noncommutative AdS_2 and subcritical values of the magnetic field the spectrum has a discrete Landau level part as well as a continuum, while for overcritical values we obtain a purely noncommutative phase consisting entirely of Landau levels.