Meissner effect from Landau problem

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 Schrödinger equation in terms of the scales ratio [Formula: see text] where the last identification is realized simply by using the Ginzburg–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, nonzero value, and therefore the theory is unstable. The supersymmetric quantum mechanics version, by completeness, is also considered.