Combinatorial theory of the semiclassical evaluation of transport moments II: Algorithmic approach for moment generating functions

TL;DR

This paper introduces an algorithmic combinatorial method to generate ribbon graphs representing classical correlations in chaotic quantum dots, enabling higher-order semiclassical calculations of transport moments.

Contribution

It develops a novel algorithmic approach to generate ribbon graphs for semiclassical transport moments, surpassing previous computational limits and suggesting patterns for higher-order terms.

Findings

Able to compute transport moments several orders higher than previous methods

Provides an expansion valid for systems with and without time reversal symmetry

Observes patterns indicating a general form for higher-order contributions

Abstract

Electronic transport through chaotic quantum dots exhibits universal behaviour which can be understood through the semiclassical approximation. Within the approximation, transport moments reduce to codifying classical correlations between scattering trajectories. These can be represented as ribbon graphs and we develop an algorithmic combinatorial method to generate all such graphs with a given genus. This provides an expansion of the linear transport moments for systems both with and without time reversal symmetry. The computational implementation is then able to progress several orders higher than previous semiclassical formulae as well as those derived from an asymptotic expansion of random matrix results. The patterns observed also suggest a general form for the higher orders.