Universal Scaling of Spectral Fluctuation Transitions for Interacting Chaotic Systems

The statistical properties of interacting strongly chaotic systems are investigated for varying interaction strength. In order to model tunable entangling interactions between such systems, we introduce a new class of random matrix transition ensembles. The nearest-neighbor-spacing distribution shows a very sensitive transition from Poisson statistics to those of random matrix theory as the interaction increases. The transition is universal and depends on a single scaling parameter only. We derive the analytic relationship between the model parameters and those of a bipartite system, with explicit results for coupled kicked rotors, a dynamical systems paradigm for interacting chaotic systems. With this relationship the spectral fluctuations for both are in perfect agreement. An accurate approximation of the nearest-neighbor-spacing distribution as a function of the transition parameter is derived using perturbation theory.

Figure 1

Nearest neighbor level-spacing distribution $P(s)$ for two coupled rotors (circles), and the RMT model (triangles). (i) For $b=0$, it is Poissonian (red line) as is the $\epsilon =0$ RMT ensemble. (ii) For $b=0.0019$, the spacing distribution is intermediate and agrees well with the corresponding RMT ensemble with $\epsilon =0.012$ and the result of a perturbation theory (blue line) for $\mathrm{\Lambda }=0.1158$. (iii) For $b=0.008$, the spacing distribution agrees well with both the RMT ensemble ($\epsilon =0.05$) and the CUE. For all cases the dimensionality of the unitary matrix is $N^2={10}^4$.

Figure 2

Universal scaling of the level-spacing distribution $P(s)$ for two coupled rotors. Three results (triangles, squares, circles) are shown for $(b,N)=(0.0025,100)$, (0.01,50), and (0.04,25). They all correspond to the same transition parameter $\mathrm{\Lambda }\approx 0.2$. The solid curve comes from a perturbation theory described in the text and the dashed curves are the Poisson and CUE distribution, respectively.

Figure 3

Variance of the nearest neighbor spacing as a function of the transition parameter. Shown are results for the coupled rotors (circles) and the RMT model (triangles), in both cases for $N=50$. The perturbation theory result is shown as a solid line (13), while the dashed line refers to a $2\times 2$ transition ensemble result [36].