The Meissner effect in the ground state of free charged Bosons in a constant magnetic field

TL;DR

This paper rigorously analyzes the Meissner effect in a model of free charged Bosons under a constant magnetic field, demonstrating exponential decay of magnetic induction in the ground state within a cylindrical geometry.

Contribution

It provides a rigorous semiclassical analysis of the Meissner effect for charged Bosons, establishing conditions for magnetic field decay in a gauge covariant setting.

Findings

Magnetic induction decays exponentially from the surface of the cylinder.

Ground state current expectation follows the London form.

Theorem on wave-function monotonicity is crucial for results.

Abstract

The model of free charged Bosons in an external constant magnetic field inside a cylinder, one of the few locally gauge covariant systems amenable to analytic treatment, is rigorously investigated in the semiclassical approximation. The model was first studied by Schafroth and is suitable for the description of quasi-bound electron pairs localized in physical space, so-called Schafroth pairs, which occur in certain compounds. Under the assumption of existence of a solution of the semiclassical problem for which the ground state (g.s.) expectation value of the current $<\vec{j}(\vec{x})>$ is of the London form, i.e., $<\vec{j}(\vec{x})> = -c |\phi_{0}(\vec{x})|^{2} \vec{A}(\vec{x})$, where c is a positive constant, $\vec{A}$ the vector potential and $\phi_{0}$ the one-particle g.s. wave-function. as well as some regularity assumptions, the magnetic induction may be proved to decay exponentially from its value on the surface of the cylinder. An important role is played by a theorem on the pointwise monotonicity of the ground state wave-function on the potential.