A solvable model of Landau quantization breakdown

TL;DR

This paper presents an exact analytical model that describes the transition from classical to quantum Hall regimes in two-dimensional electron gases by solving a non-rotationally invariant harmonic potential using 4D phase space quantization.

Contribution

It introduces a solvable model capturing the three key phases of Landau quantization breakdown with a unified analytical approach.

Findings

Identifies three distinct phases of Landau quantization as magnetic field varies.

Provides an exact quantum solution for a non-rotationally invariant potential.

Unifies classical, quantum oscillations, and quantum Hall regimes under one theory.

Abstract

Physics of two-dimensional electron gases under perpendicular magnetic field often displays three distinct stages when increasing the field amplitude: a low field regime with classical magnetotransport, followed at intermediate field by a Shubnikov-de Haas phase where the transport coefficients present quantum oscillations, and, ultimately, the emergence at high field of the quantum Hall effect with perfect quantization of the Hall resistance. A rigorous demonstration of this general paradigm is still limited by the difficulty in solving models of quantum Hall bars with macroscopic lateral dimensions and smooth disorder. We propose here the exact solution of a simple model exhibiting similarly two sharp transitions that are triggered by the competition of cyclotron motion and potential-induced drift. As a function of increasing magnetic field, one observes indeed three distinct phases showing respectively fully broken, partially smeared, or perfect Landau level quantization. This model is based on a non-rotationally invariant, inverted two-dimensional harmonic potential, from which a full quantum solution is obtained using 4D phase space quantization. The developed formalism unifies all three possible regimes under a single analytical theory, as well as arbitrary quadratic potentials, for all magnetic field values.