Spectral Properties of Quantum Circulant Graphs

TL;DR

This paper introduces quantum circulant graphs, analyzing their spectral properties, and reveals how symmetry and edge lengths influence spectral statistics, including connections to random matrix theory and applications to spectral zeta functions.

Contribution

It develops a new model for quantum circulant graphs, deriving secular equations and analyzing spectral statistics under different symmetry conditions, extending understanding beyond star and Dirac rose graphs.

Findings

Spectral statistics follow GOE for incommensurate edge lengths.

Spectra decompose into irreducible representations when symmetry is present.

Intermediate spectral statistics are observed in symmetric cases.

Abstract

We introduce a new model for investigating spectral properties of quantum graphs, a quantum circulant graph. Circulant graphs are the Cayley graphs of cyclic groups. Quantum circulant graphs with standard vertex conditions maintain important features of the prototypical quantum star graph model. In particular, we show the spectrum is encoded in a secular equation with similar features. The secular equation of a quantum circulant graph takes two forms depending on whether the edge lengths respect the cyclic symmetry of the graph. When all the edge lengths are incommensurate, the spectral statistics correspond to those of random matrices from the Gaussian Orthogonal Ensemble according to the conjecture of Bohigas, Giannoni and Schmit. When the edge lengths respect the cyclic symmetry the spectrum decomposes into subspectra whose corresponding eigenfunctions transform according to irreducible representations of the cyclic group. We show that the subspectra exhibit intermediate spectral statistics and analyze the small and large parameter asymptotics of the two-point correlation function, applying techniques developed from star graphs. The particular form of the intermediate statistics differs from that seen for star graphs or Dirac rose graphs. As a further application, we show how the secular equations can be used to obtain spectral zeta functions using a contour integral technique. Results for the spectral determinant and vacuum energy of circulant graphs are obtained from the zeta functions.