Meissner Effect from Landau Problem

TL;DR

This paper explores the Landau problem with inhomogeneous magnetic fields, revealing a Meissner effect and vortex formation, and connects these phenomena to superconductivity and supersymmetric quantum mechanics.

Contribution

It demonstrates the emergence of the Meissner effect and vortices in the Landau problem with non-homogeneous magnetic fields, linking to superconductivity and supersymmetric quantum mechanics.

Findings

Landau problem with inhomogeneous fields shows Meissner effect

Vortices emerge unless two-body interactions are added

Equations are valid in the superconducting regime

Abstract

The Landau problem for inhomogeneous magnetic fields is examined in a very general context and several interesting analogies with the Nielsen-Olesen vortices are established. Firstly we show that the Landau problem with non-homogeneous magnetic fields exhibits Meissner effect that is unstable unless two-body interactions are added and vortices emerge. Using the scaling freedom we can write the Schr\"odinger equation in terms of the scales ratio $\kappa ={ E}/{ m }\propto 1- T/T_c $ where the last identification is realised simply by using the Gizburg-Landau theory. We find our equations are valid in the superconducting regime, and it is not possible for the Cooper pairs amplitude to reach to a constant, non-zero value, and therefore the theory is unstable. The supersymmetric quantum mechanics version, by completeness, is also considered.