Ballistic Quantum Dots with Disorder and Interactions: A numerical study on the Robnik-Berry billiard

TL;DR

This numerical study of the Robnik-Berry billiard tests previous theoretical predictions about electron interactions in ballistic quantum dots, confirming most assumptions but noting some fluctuations in matrix elements.

Contribution

The paper numerically validates earlier theoretical assumptions about interactions in ballistic quantum dots using the Robnik-Berry billiard model.

Findings

Most predictions based on earlier work are confirmed.

Fluctuations of certain matrix elements differ from predictions.

Results support the qualitative and semi-quantitative validity of previous theories.

Abstract

In previous work we have found a regime in ballistic quantum dots where interelectron interactions can be treated asymptotically exactly as the Thouless number $g$ of the dot becomes very large. However, this work depends on some assumptions concerning the renormalization group and various properties of the dot obeying Random Matrix Theory predictions at scales of the order of the Thouless energy. In this work we test the validity of those assumptions by considering a particular ballistic dot, the Robnik-Berry billiard, numerically. We find that almost all of our predictions based on the earlier work are borne out, with the exception of fluctuations of certain matrix elements of interaction operators. We conclude that, at least in the Robnik-Berry billiard, one can trust the results of our previous work at a qualitative and semi-quantitative level.