Surmise for random matrices' level spacing distributions beyond nearest-neighbors

TL;DR

The paper extends the Wigner surmise to $k$-th nearest-neighbor spacings in Gaussian random-matrix ensembles by correcting the exponent using the exact $k$NN variance, yielding a surmise $P^{(k)}(s) \approx C_{\tilde{\alpha}} \, s^{\tilde{\alpha}} e^{-A_{\tilde{\alpha}} s^2}$ with $\tilde{\alpha}$ calibrated to $\Delta^{(k)}_\beta$. This approach reproduces the known variance across GOE, GUE, and GSE and outperforms the traditional generalized surmise across a range of $k$, as validated against large random matrices. The authors also show a Gaussian limit for large $k$, and demonstrate the utility of the corrected surmise on the XXZ spin chain with random disorder to probe the chaos-to-integrability transition more finely than scalar indicators. Overall, the work provides a practical, accurate model for long-range spectral correlations and a quantitative tool for studying many-body quantum chaos.

Abstract

Correlations between energy levels can help distinguish whether a many-body system is of integrable or chaotic nature. The study of short-range and long-range spectral correlations generally involves quantities which are very different, unless one uses the $k$-th nearest neighbor ($k$NN) level spacing distributions. For nearest-neighbor (NN) spectral spacings, the distribution in random matrices is well captured by the Wigner surmise. This well-known approximation, derived exactly for a 2$\times$2 matrix, is simple and satisfactorily describes the NN spacings of larger matrices. There have been attempts in the literature to generalize Wigner's surmise to further away neighbors. However, as we show, the current proposal in the literature fails to accurately capture numerical data. Using the known variance of the distributions from random matrix theory, we propose a corrected surmise for the $k$NN spectral distributions. This surmise better characterizes spectral correlations while retaining the simplicity of Wigner's surmise. We test the predictions against numerical results and show that the corrected surmise is systematically more accurate at capturing data from random matrices. Using the XXZ spin chain with random on-site disorder, we illustrate how these results can be used as a refined probe of many-body quantum chaos for both short- and long-range spectral correlations.

Figures (10)

• Figure 1: The variance and mean (inset) of the $k$th neighbor spacing distributions for the GOE, GUE, and GSE, for $k\geq 1$. The red dots show the analytical values using the new surmise, i.e. the exponent Eq. \ref{['alpha_new']} in the expression for the variance Eq. \ref{['variance']}. The blue dots show analytical values for the old surmise, i.e. using the exponent Eq. \ref{['alpha_old']} in Eq. \ref{['variance']}. The grey crosses show numerical data for 1000 realizations of random matrices of dimension $N=1000$ for the GOE and GUE and $N=2000$ for the GSE. In all cases, the numerical mean for the unfolded spectrum satisfies $\langle s^{(k)} \rangle=k$.
• Figure 2: The standard deviation (goodness of fit), $\sigma$, defined in Eq. \ref{['SD']}, for $k\geq 2$ computed for the old surmise (blue), the new surmise (red) and the Gaussian surmise (gold) against numerical data of 1000 realizations of random matrices of dimension $N=1000$ for the GOE and GUE and $N=2000$ for the GSE. The new surmise Eq. \ref{['Pk_new']} systematically better captures the numerical data in the three ensembles.
• Figure 3: The $k$NN distributions for a few values of $k$, for the GUE. The histograms show the numerical data for 1000 realizations of $N=1000$ matrices taken from the GUE. The top row shows the old surmise (blue curves), while the bottom row shows the new surmise (red curves). The standard deviation goodness-of-fit, $\sigma$, between the analytical distributions and the numerical data is shown in units of $10^{-2}$. See Appendix \ref{['App:GOE_GSE']} for similar data for the GOE and GSE.
• Figure 4: Results for the XXZ spin chain with random on-site magnetic fields. (Left) The goodness-of-fit, $\sigma$, for $W=1, 1.25, 1.5, 1.75$ and $2$, compared with the $k$NN GOE distributions for $2\leq k \leq 35$. The dashed red line at $\sigma=0.3\times 10^{-2}$ is the benchmark value discussed in the main text; the connecting lines are shown as a guide to the eye. (Middle) The variance of the $k$NN distributions for the disorder strengths $W$ listed in the legend (log-scale on the $y$-axis). For reference we added the variances Eq. \ref{['Delta_RMT']} for the GOE (red dots) and the variances Eq. \ref{['Delta_Poisson']} for the 1d Poisson point process (black dots). The number variance is plotted as a continuous line. (Right) The difference between the Number Variance and the variances of the $k$NN distributions (crosses; the connecting lines are shown as a guide to the eye). The value of $1/6$ is plotted as a horizontal red line. The full distribution histograms are shown for a few values of $k$ in Figure \ref{['fig:XXZ_kNN']}.
• Figure 5: Numerical data for the $k$NN distributions for the XXZ spin chain with random on-site magnetic fields (step histograms) For $W=1,2,3,4,5$ and $20$. The red smooth curve shows the corrected surmise Eq. \ref{['Pk_new']} and the black smooth curve shows the $k$NN distributions for the 1d Poisson point process Eq. \ref{['Pk_Poisson']}. We see a refined picture of how the spectral statistics change from chaos to integrability at various spectral ranges.