Localization by interference: Square billiard with a magnetic flux

TL;DR

This paper investigates the quantum eigenstates of a square billiard in a magnetic field, revealing new localized states caused by phase interference that significantly influence magnetic response, with explicit formulas derived for energy levels and wavefunctions.

Contribution

It introduces the discovery of previously unnoticed localized states in a square billiard with magnetic flux, explaining their origin and impact.

Findings

Localized states due to phase interference are identified.

Explicit formulas for energy levels and wavefunctions are provided.

Localized states dominate magnetic response in certain conditions.

Abstract

Eigenstates and energy levels of a square quantum billiard in a magnetic field, or with an Aharonov-Bohm flux line, are found in quasiclassical approximation, that is, for high enough energy. Explicit formulas for the energy levels and wavefunctions are found. There are localized states, never before noticed in this well studied problem, whose localization is due to phase interference, even though there is no or negligible classical effect of the magnetic field. These and related states account almost entirely for the magnetic response in certain temperature ranges, and thus have a bearing on the experiments of Levy et al.