Diagrammatic Analysis of the Unitary Group for Double Barrier Ballistic Cavities: Equivalence with Circuit Theory

TL;DR

This paper establishes an analytical connection between diagrammatic techniques for unitary group averages and Nazarov's circuit theory in the context of double-barrier ballistic cavities, simplifying the analysis of their statistical properties.

Contribution

It demonstrates the equivalence between complex diagrammatic equations and simpler polynomial equations derived from circuit theory for chaotic quantum dots.

Findings

Equivalence between diagrammatic and circuit theory approaches.

Simplification of complex equations to polynomial form.

Potential for streamlined analysis of quantum dot statistics.

Abstract

We derive a set of coupled non-linear algebraic equations for the asymptotics of the Poisson kernel distribution describing the statistical properties of a two-terminal double-barrier chaotic billiard (or ballistic quantum dot). The equations are calculated from a diagrammatic technique for performing averages over the unitary group, proposed by Brouwer and Beenakker [J. Math. Phys. 37, 4904 (1996)]. We give strong analytical evidences that these equations are equivalent to a much simpler polynomial equation calculated from a recent extension of Nazarov's circuit theory [A. M. S. Macedo, Phys. Rev. B 66, 033306 (2002)]. These results offer interesting perspectives for further developments in the field via the direct conversion of one approach into the other.