Periodic Orbit Theory of the circular billiard in homogeneous magnetic fields

TL;DR

This paper develops a semiclassical trace formula for a circular quantum dot in a magnetic field, incorporating boundary and grazing orbit effects, to accurately predict level densities and shell structures.

Contribution

It extends Gutzwiller theory to include boundary and grazing orbit effects, improving semiclassical predictions for quantum dots in magnetic fields.

Findings

Grazing orbits are the dominant correction to semiclassical results.

Replacing the Maslov index with a quantum reflection phase improves accuracy.

The main shell structure features are explained by one or two classical orbits.

Abstract

We present a semiclassical description of the level density of a two-dimensional circular quantum dot in a homogeneous magnetic field. We model the total potential (including electron-electron interaction) of the dot containing many electrons by a circular billiard, i.e., a hard-wall potential. Using the extended approach of the Gutzwiller theory developed by Creagh and Littlejohn, we derive an analytic semiclassical trace formula. For its numerical evaluation we use a generalization of the common Gaussian smoothing technique. In strong fields orbit bifurcations, boundary effects (grazing orbits) and diffractive effects (creeping orbits) come into play, and the comparison with the exact quantum mechanical result shows major deviations. We show that the dominant corrections stem from grazing orbits, the other effects being much less important. We implement the boundary effects, replacing the Maslov index by a quantum-mechanical reflection phase, and obtain a good agreement between the semiclassical and the quantum result for all field strengths. With this description, we are able to explain the main features of the gross-shell structure in terms of just one or two classical periodic orbits.