Quantum mechanics and quantum Hall effect on Riemann surfaces

TL;DR

This paper explores quantum mechanics and the quantum Hall effect on Riemann surfaces, constructing explicit wave functions for ground and Landau level states, and analyzing their degeneracies using advanced mathematical tools.

Contribution

It provides explicit constructions of wave functions for quantum states on Riemann surfaces and analyzes their degeneracies with the Riemann-Roch theorem, advancing understanding of quantum Hall physics in curved spaces.

Findings

Explicit wave functions for ground states and Landau levels on Riemann surfaces.

Degeneracy of Landau levels derived via Riemann-Roch theorem.

Construction and analysis of Laughlin wave functions on Riemann surfaces.

Abstract

The quantum mechanics of a system of charged particles interacting with a magnetic field on Riemann surfaces is studied. We explicitly construct the wave functions of ground states in the case of a metric proportional to the Chern form of the $\theta$-bundle, and the wave functions of the Landau levels in the case of the the Poincar{\' e} metric. The degeneracy of the the Landau levels is obtained by using the Riemann-Roch theorem. Then we construct the Laughlin wave function on Riemann surfaces and discuss the mathematical structure hidden in the Laughlin wave function. Moreover the degeneracy of the Laughlin states is also discussed.