标题： 系综平均的平均场多体能级密度：可积与混沌单粒子动力学的指示器 显示英文标题

标题： Ensemble-averaged mean-field many-body level density: an indicator of integrable versus chaotic single-particle dynamics

摘要： 根据量子混沌范式，系统的经典动力学性质（可积或混沌）在量子谱的波动中普遍体现。 然而，由于平均场极限下的多体谱由独立的单粒子能级组成，其谱波动始终表现出泊松行为，因此不能用来区分潜在的混沌与可积的单粒子动力学。 我们证明，这种区分可以从固定能量下的平均多体能级密度及其在代表不同单粒子动力学类型的系综中的方差中揭示出来。 这与能量平均的平均能级密度（给定系统的）形成强烈对比，后者被认为不包含此类信息，并通常被去除以专注于普遍特征。 为了支持我们的主张，我们系统地分析了单粒子能级相关性的作用，这些相关性通过泊松和随机矩阵统计（不同对称类）进入系综平均态密度及其方差，并对比了玻色子和费米子多体系统。 我们的分析研究，以及对$N \ge 5$个粒子系统的广泛数值模拟，一致显示出显著差异（对于费米子可达一个数量级，对于玻色子甚至更大），这取决于底层动力学的性质。 值得注意的是，在费米子情况下，泊松型单粒子能级波动恰好抵消了不可区分性的影响，使得平均多体谱密度等于（托马斯-费米）体积项。 我们进一步强调了平均能级密度及其方差作为总能量$E$和激发能$Q$的函数之间的差异。 显示英文摘要

摘要： According to the quantum chaos paradigm, the nature of a system's classical dynamics, whether integrable or chaotic, is universally reflected in the fluctuations of its quantum spectrum. However, since many-body spectra in the mean field limit are composed of independent single-particle energy levels, their spectral fluctuations always display Poissonian behavior and hence cannot be used to distinguish underlying chaotic from integrable single-particle dynamics. We demonstrate that this distinction can, instead, be revealed from the mean many-body level density (at fixed energy) and its variance after averaging over ensembles representing different types of single-particle dynamics. This is in strong contrast to the energy-averaged mean level density (of a given system) that is assumed not to carry such information and is routinely removed to focus on universal signatures. To support our claim we systematically analyze the role of single-particle level correlations, that enter through Poisson and random matrix statistics (of various symmetry classes) into the ensemble-averaged density of states and its variance, contrasting bosonic and fermionic many-body systems. Our analytical study, together with extensive numerical simulations for systems with $N \ge 5$ particles consistently reveal significant differences (up to an order of magnitude for fermions and even larger for bosons) in the mean many-body level densities, depending on the nature of the underlying dynamics. Notably, in the fermionic case Poisson-type single-particle level fluctuations precisely cancel contributions from indistinguishability, such that the average many-body spectral density equals the (Thomas-Fermi) volume term. We further highlight the difference between the mean level density and its variance as functions of the total energy $E$ and the excitation energy $Q$.