BCS Theory in the Weak Magnetic Field Regime for Systems with Nonzero Flux and Exponential Estimates on the Adiabatic Theorem in Extended Quantum Lattice Systems

TL;DR

This thesis analyzes BCS theory in superconductors under weak magnetic fields with nonzero flux, establishing Ginzburg--Landau limits, and explores exponential estimates for the adiabatic theorem in quantum lattice systems.

Contribution

It extends BCS theory to include non-vanishing magnetic flux and provides new exponential estimates for the adiabatic theorem in extended quantum lattice systems.

Findings

Superconductors are described by Ginzburg--Landau theory in the weak magnetic field limit.

Allowing non-zero magnetic flux through the lattice unit cell.

Provides exponential estimates for the adiabatic theorem in quantum lattice systems.

Abstract

In the main part of this PhD thesis, we consider a periodically realized microscopic superconductor described by BCS theory, which is subject to external electromagnetic fields. We show that the superconductor is properly described by Ginzburg--Landau theory in the macroscopic and weak magnetic field limit. The main novelty of our results is to allow for a non-vanishing magnetic flux through the unit cell of the lattice of periodicity. These main results are supplemented by various unpublished notes in the field of BCS theory. Furthermore, we preface the presentation of these results with a comprehensive introduction suitable for master's or PhD students. Thereby, we hope to contribute to filling the gap of missing introductory literature in the field. The thesis comprises a second topic, in which we provide ideas for setting up quantum lattice systems in order to prove exponential estimates for the adiabatic theorem. These notes are the result of studies in this field, which have been conducted during a research stay at the University of British Columbia (UBC) in Vancouver, Canada.