Lack of diamagnetism and the Little-Parks effect

TL;DR

This paper rigorously justifies the Little-Parks effect, showing that superconductivity loss under magnetic fields is not at a single value and can oscillate with non-uniform fields, based on spectral theory analysis.

Contribution

It provides a rigorous mathematical proof of the Little-Parks effect and demonstrates oscillations in superconductivity for non-uniform magnetic fields in specific sample shapes.

Findings

Superconductivity persists over non-interval sets of magnetic field strengths.

Oscillations in superconductivity can occur repeatedly at large Ginzburg--Landau parameters.

Ground state energy of magnetic Schrödinger operators is not monotone in field strength.

Abstract

When a superconducting sample is submitted to a sufficiently strong external magnetic field, the superconductivity of the material is lost. In this paper we prove that this effect does not, in general, take place at a unique value of the external magnetic field strength. Indeed, for a sample in the shape of a narrow annulus the set of magnetic field strengths for which the sample is superconducting is not an interval. This is a rigorous justification of the Little-Parks effect. We also show that the same oscillation effect can happen for disc-shaped samples if the external magnetic field is non-uniform. In this case the oscillations can even occur repeatedly along arbitrarily large values of the Ginzburg--Landau parameter $\kappa$. The analysis is based on an understanding of the underlying spectral theory for a magnetic Schr\"{o}dinger operator. It is shown that the ground state energy of such an operator is not in general a monotone function of the intensity of the field, even in the limit of strong fields.