Finite-range Coulomb gas models of banded random matrices and quantum kicked rotors

Dyson demonstrated an equivalence between infinite-range Coulomb gas models and classical random matrix ensembles for the study of eigenvalue statistics. We introduce finite-range Coulomb gas (FRCG) models via a Brownian matrix process, and study them analytically and by Monte Carlo simulations. These models yield new universality classes, and provide a theoretical framework for the study of banded random matrices (BRMs) and quantum kicked rotors (QKRs). We demonstrate that, for a BRM of bandwidth $b$ and a QKR of chaos parameter $\alpha$, the appropriate FRCG model has the effective range $d=b^2/N={\alpha }^2/N$, for large $N$ matrix dimensionality. As $d$ increases, there is a transition from Poisson to classical random matrix statistics.

Figure 1

Spacing density $p_5\left(s\right)$ for $d=2$ for (a) $\beta =1$ and (b) $\beta =2$.

Figure 2

Spacing density $p_k\left(s\right)$ for $k=0,5$. Left and right panels are for $\beta =1$ and $\beta =2$, respectively. (a), (b), (e), (f) correspond to $d=2$; (c), (d), (g), (h) correspond to $d=5$.

Figure 3

Spacing density $p_5\left(s\right)$ for $d=2.5$ for (a) $\beta =1$ and (b) $\beta =2$.

Figure 4

Spacing variance for $d=0$ $(\mathrm{Poisson}),1,2,5,10,25,\infty$ (GE) (from top to bottom) for (a) $\beta =1$ and (b) $\beta =2$. The MC results have been connected by a line for visual convenience.

Figure 5

Plot of $\mathrm{var}\left(L\right)/\mathrm{var}\left(1\right)$ vs $L$ and scaled variable $L/d$, in (a) and (b), respectively, both for QKR. The inset in (a) is for large $\alpha$ such that $d\simeq O\left(N\right)$. In Fig. 4, $d=1,5,10,25$ (from innermost to outermost). We have taken $N=1000$.