Intermediate statistics for a system with symplectic symmetry: the Dirac rose graph

TL;DR

This paper investigates the spectral statistics of the Dirac operator on a rose-shaped graph, deriving a secular equation and approximations for spectral correlations, marking the first example of intermediate statistics in the symplectic class.

Contribution

It introduces a secular equation for the Dirac rose graph and provides the first example of intermediate spectral statistics for the symplectic symmetry class.

Findings

Derived a secular equation for the Dirac rose graph.

Provided spectral pair correlation approximations at large and small spacings.

Confirmed predictions through numerical calculations.

Abstract

We study the spectral statistics of the Dirac operator on a rose-shaped graph---a graph with a single vertex and all bonds connected at both ends to the vertex. We formulate a secular equation that generically determines the eigenvalues of the Dirac rose graph, which is seen to generalise the secular equation for a star graph with Neumann boundary conditions. We derive approximations to the spectral pair correlation function at large and small values of spectral spacings, in the limit as the number of bonds approaches infinity, and compare these predictions with results of numerical calculations. Our results represent the first example of intermediate statistics from the symplectic symmetry class.