Random matrix theory of integrability-to-chaos transition

Abstract

The statistics of gaps between quantum energy levels is a hallmark criterion in quantum chaos and quantum integrability studies. The relevant distributions corresponding to exactly integrable vs. fully chaotic systems are universal and described by the Poisson vs. Wigner-Dyson curves. In the transitional regime between integrability and chaos, the distributions are much less universal and have not been understood quantitatively until now. We point out that the relevant statistics that controls these distributions is that of the matrix elements of the nonintegrable perturbation Hamiltonian in the energy eigenbasis of the unperturbed integrable system. With this insight, we formulate a simple random matrix ensemble that correctly reproduces the level spacing distributions in a variety of test systems. For the distribution of matrix elements appearing in our construction, we furthermore discover surprising universal features: across a variety of physical systems with diverse degrees of freedom, these distributions are dominated by simple power laws.

Figures (9)

• Figure 1: The transition in level spacing distribution from Poisson to WD at four different values of $r$ for the spin chain model (\ref{['eq: H0 spin chain']}-\ref{['eq: H1 spin chain']}) at $L = 16$ plotted as filled circles, compared with the joint distribution of spacings for 50 realizations of the random matrix ensemble (\ref{['eq: RM ensemble']}) in dashed magenta. The Hilbert space dimension is $D=32896$. As a matter of comparison, we also display the Poisson distribution \ref{['eq: Poisson distribution']} in red, the Wigner surmise \ref{['eq: Wigner Dyson surmise']} in blue, the level spacing statistics of the RP model, based on 50 realizations at $D= 32896$, at the same values of $r$ in dash-dotted black, and the Brody distribution as a dotted black curve, with $\beta$ found by fitting \ref{['eq: Brody distribution']} to the physical level spacing distribution.
• Figure 2: Similar to Fig. \ref{['fig: Ising histograms']}, but now for the perturbed integrable QRS model \ref{['eq: truncated szego C']} at $(N,M)=(37,37)$ with $D=21637$.
• Figure 3: The probability density on a log-log scale for the magnitude of offdiagonal elements of the random perturbations $H_1$ in the eigenbasis of $H_0$ for (left) the spin chain model (\ref{['eq: H0 spin chain']}-\ref{['eq: H1 spin chain']}) at $L=16$ and (right) the perturbed integrable QRS model \ref{['eq: truncated szego C']} at $(N,M)=(37,37)$. These distributions were used as $P_{\mathcal{M}}$ in our random matrix model approach to the crossover statistics. The log-log histograms reveal a power law regime (magenta) that extends over almost 10 orders of magnitude in both cases. The shape of the distributions outside of this regime plays no important role for the crossover statistics.
• Figure 4: Distribution of $r_n$ for the spin chain model (filled circles), compared to the prediction of our random model (magenta dashed) and the RP model (black dash-dotted). Note that $r$ in the legends stands for the average value of the $r$-ratios.
• Figure 5: Distribution of $r_n$ for QRS.