Universal spectral statistics in Wigner-Dyson, chiral, and Andreev star graphs. I. Construction and numerical results

In a series of two papers we investigate the universal spectral statistics of chaotic quantum systems in the ten known symmetry classes of quantum mechanics. In this first paper we focus on the construction of appropriate ensembles of star graphs in the ten symmetry classes. A generalization of the Bohigas-Giannoni-Schmit conjecture is given that covers all these symmetry classes. The conjecture is supported by numerical results that demonstrate the fidelity of the spectral statistics of star graphs to the corresponding Gaussian random-matrix theories.

Figure 1

Sketch of a star graph with five bonds of equal length.

Figure 2

Second-order form factor $K_2\left(\tau \right)$ for star graphs in the Wigner-Dyson ensembles averaged over 10 000 realizations of with $B=100$ bonds [additional time average over an interval of length $8\left(g∕MB\right)$] (a). Symmetry class $A\text{(GUE)}$, (b) symmetry class $A\mathrm{I}\text{(GOE)}$, (c) symmetry class $A\text{II}\text{(GSE)}$. Dashed lines: prediction by Gaussian random-matrix theory. Full lines: numerically calculated form factor for graphs. Inserts: discrete time form factor $K_{2,n}$ as function of $\tau =n\left(g∕MB\right)$ for the class $A$ and $A\mathrm{I}$ graphs. For the class $A\text{II}$ graphs $K_{2,n}$ vanishes for odd $n$—the insert shows $K_{2,n}∕2$ for even $n$.

Figure 3

First-order form factor for Andreev star graphs in the symmetry classes $C$ (a) and $C\mathrm{I}$ (b) averaged over $\mathrm{10\hspace{0.17em}000}$ realizations with $B=100$ bonds [additional time average over an interval of length $8(g∕MB)$]. Dashed lines: prediction by Gaussian random-matrix theory. Full lines: numerically calculated form factor for graphs. Inserts: discrete time form factor $K_{1,n}∕2$ as function of $\tau =n\left(g∕MB\right)$ for even $n$ ($K_{1,n}$—this vanishes by construction for odd $n$).

Figure 4

The first-order form factor for Andreev star graphs in the classes $D$ (a) and $D\text{III}$ (b)—see Fig. 3 for details.

Figure 5

The first-order form factor for chiral star graphs in the symmetry classes $A\text{III}$ (a), $BD\mathrm{I}$ (b), and $C\text{II}$ (c)—see Fig. 3 for details.