可积与非可积量子哈密顿量的统计特征
Statistical Signatures of Integrable and Non-Integrable Quantum Hamiltonians
AI总结：
本文提出了一种基于概率的统计框架，通过蒙特卡洛退相干和k步间隙分布分析，区分可积与非可积量子哈密顿量，验证了其在不同结构上的有效性。
摘要：
可积性是经典力学中的基石，具有明确的含义。然而，将其扩展到量子系统时，仍然显得微妙且未解决。特别是，决定一个量子哈密顿量——仅作为矩阵来看——是否定义了一个可积系统，远非显而易见，但对理解非平衡动力学、能谱相关性和许多体物理中的关联函数至关重要。我们开发了一种统计框架，从概率的角度出发来探讨量子可积性。一个关键的观察是，可积性要求消失的能量间隙具有有限的概率。在此基础上，我们提出了一种两步协议来区分可积和非可积哈密顿量。首先，我们应用系统性的蒙特卡洛退相干，这会指数级压缩希尔伯特空间，并揭示能级间距是否接近泊松统计或保持混合。该退相干的终止点表明了谱的统计特性。其次，我们分析k步间隙分布，这会加强泊松和混合统计之间的区分。我们的方法适用于任何有限大小的哈密顿量，无论其结构是否包含几个块或指数碎片化的希尔伯特空间。作为基准，我们将在由置换群S_N构建的量子哈密顿量上实施该协议，证明了其有效性和普遍性。
英文摘要：
Integrability is a cornerstone of classical mechanics, where it has a precise meaning. Extending this notion to quantum systems, however, remains subtle and unresolved. In particular, deciding whether a quantum Hamiltonian - viewed simply as a matrix - defines an integrable system is far from obvious, yet crucial for understanding non-equilibrium dynamics, spectral correlations, and correlation functions in many-body physics. We develop a statistical framework that approaches quantum integrability from a probabilistic standpoint. A key observation is that integrability requires a finite probability of vanishing energy gaps. Building on this, we propose a two-step protocol to distinguish integrable from non-integrable Hamiltonians. First, we apply a systematic Monte Carlo decimation of the spectrum, which exponentially compresses the Hilbert space and reveals whether level spacings approach Poisson statistics or remain mixed. The termination point of this decimation indicates the statistical character of the spectrum. Second, we analyze $k$-step gap distributions, which sharpen the distinction between Poisson and mixed statistics. Our procedure applies to Hamiltonians of any finite size, independent of whether their structure involves a few blocks or an exponentially fragmented Hilbert space. As a benchmark, we implement the protocol on quantum Hamiltonians built from the permutation group $\mathcal{S}_N$, demonstrating both its effectiveness and generality.
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