Combinatorial theory of the semiclassical evaluation of transport moments I: Equivalence with the random matrix approach

TL;DR

This paper proves that semiclassical and random matrix approaches to quantum transport moments are universally equivalent by developing a combinatorial interpretation involving ribbon graphs and primitive factorizations.

Contribution

It establishes a universal equivalence between semiclassical and random matrix evaluations of transport moments using combinatorial ribbon graph methods.

Findings

Semiclassical and random matrix evaluations are shown to be equivalent.

Ribbon graphs are used to interpret trajectory sets combinatorially.

The approach applies to systems with and without time reversal symmetry.

Abstract

To study electronic transport through chaotic quantum dots, there are two main theoretical approachs. One involves substituting the quantum system with a random scattering matrix and performing appropriate ensemble averaging. The other treats the transport in the semiclassical approximation and studies correlations among sets of classical trajectories. There are established evaluation procedures within the semiclassical evaluation that, for several linear and non-linear transport moments to which they were applied, have always resulted in the agreement with random matrix predictions. We prove that this agreement is universal: any semiclassical evaluation within the accepted procedures is equivalent to the evaluation within random matrix theory. The equivalence is shown by developing a combinatorial interpretation of the trajectory sets as ribbon graphs (maps) with certain properties and exhibiting systematic cancellations among their contributions. Remaining trajectory sets can be identified with primitive (palindromic) factorisations whose number gives the coefficients in the corresponding expansion of the moments of random matrices. The equivalence is proved for systems with and without time reversal symmetry.