标题： 对随机矩阵的相邻间距分布之外的推测 显示英文标题

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

摘要： 能量水平之间的相关性有助于区分多体系统是可积的还是混沌的。 研究短程和长程光谱相关性通常涉及量，这些量在没有使用$k$-邻近（$k$NN）水平间距分布的情况下非常不同。 对于最近邻（NN）光谱间距，随机矩阵中的分布由威格纳猜测很好地描述。 这个众所周知的近似值，精确地来源于 2$\times$2 矩阵，简单且令人满意地描述了更大矩阵的 NN 间距。 文献中曾尝试将威格纳猜测推广到更远的邻居。 然而，正如我们所展示的，文献中的当前提议并不能准确捕捉数值数据。 利用随机矩阵理论中分布的已知方差，我们提出了一个修正的猜测用于$k$NN 光谱分布。 这个猜测更好地表征了光谱相关性，同时保留了威格纳猜测的简洁性。 我们将预测与数值结果进行测试，并表明修正的猜测在捕捉随机矩阵的数据方面系统性地更好。 使用具有随机位点无序的 XXZ 自旋链，我们说明了这些结果如何可以作为对多体量子混沌的精细探测工具，用于短程和长程光谱相关性。 显示英文摘要

摘要： 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 distribution. 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 does not 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 better 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.