Spectral statistics of nearly unidirectional quantum graphs

TL;DR

This paper investigates how breaking unidirectionality in quantum graphs affects their spectral statistics, deriving an analytic model that matches numerical results for certain graph classes.

Contribution

It introduces a random matrix model to analytically describe spectral statistics of nearly unidirectional quantum graphs with broken unidirectionality.

Findings

Analytic expression for nearest neighbour energy level distribution.

Excellent agreement with graphs having uniform eigenfunction distribution.

Significant deviations for graphs with strong scarring.

Abstract

The energy levels of a quantum graph with time reversal symmetry and unidirectional classical dynamics are doubly degenerate and obey the spectral statistics of the Gaussian Unitary Ensemble. These degeneracies, however, are lifted when the unidirectionality is broken in one of the graph's vertices by a singular perturbation. Based on a Random Matrix model we derive an analytic expression for the nearest neighbour distribution between energy levels of such systems. As we demonstrate the result agrees excellently with the actual statistics for graphs with a uniform distribution of eigenfunctions. Yet, it exhibits quite substantial deviations for classes of graphs which show strong scarring.