Meissner Effect from Landau Problem

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 κ = E/m 1-T/Tc 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. © 2018 World Scientific Publishing Company.