Landau Level Ground--State Degeneracy, and Its Relevance for a General Quantization Procedure

TL;DR

This paper investigates the degeneracy of Landau levels on curved surfaces with nonuniform magnetic fields, proposing a new, coordinate-invariant quantization method that extends traditional approaches and aids in deriving semiclassical Hall current formulas.

Contribution

It introduces a novel, general quantization procedure invariant under coordinate transformations, applicable to complex geometries and magnetic fields, and connects this to Landau level degeneracy and Hall effect calculations.

Findings

Lowest Landau level remains degenerate under general conditions.

Proposed quantization method is invariant under coordinate changes.

Derived semiclassical formulas for Hall current at filling factor one.

Abstract

The quantum dynamics of a two-dimensional charged spin $1/2$ particle is studied for general, symmetry--free curved surfaces and general, nonuniform magnetic fields that are, when different from zero, orthogonal to the defining two surface. Although higher Landau levels generally lose their degeneracy under such general conditions, the lowest Landau level, the ground state, remains degenerate. Previous discussions of this problem have had less generality and/or used supersymmetry, or else have appealed to very general mathematical theorems from differential geometry. In contrast our discussion relies on simple and standard quantum mechanical concepts. The mathematical similarity of the physical problem at hand and that of a phase-space path integral quantization scheme of a general classical system is emphasized. Adopting this analogy in the general case leads to a general quantization procedure that is invariant under general coordinate transformations-- completely unlike any of the conventional quantization prescriptions -- and therefore generalizes the concept of quantization to new and hitherto inaccesible situations. In a complementary fashion , the so-obtained picture of general quantization helps to derive useful semiclassical formulas for the Hall current in the case of a filling factor equal to one for a general surface and magnetic field.