AI 中文总结
研究广义罗森茨维格 - 波特模型分形相中的能级统计，利用自由概率论和 replica 方法计算大系统尺寸极限下的全计数统计，通过数值对角化和大偏差算法验证预测，并与量子随机能量模型测量结果对比。
AI 中文摘要
罗森茨维格 - 波特（RP）随机矩阵系综已成为量子多体系统中从可积到混沌转变的最小模型。其相图有一个具有分形本征态的区域，在完全局域化和完全非局域化状态之间展现出中间光谱和局域化特性。本文探索了RP模型的几种推广形式，并在表征转变的 Thouless 能量\(E_T\)尺度下确定其能级统计。利用自由概率论和 replica 方法的工具，计算了大系统尺寸极限下的全计数统计，表明在\(E_T\)附近它具有简单通用的标度形式，所有模型变体都相同。用大样本的精确数值对角化和能解析低至\(10^{-40}\)概率的全计数统计的大偏差算法验证了分析预测。还将预测与量子随机能量模型的测量结果进行了对比。
英文摘要
The Rosenzweig--Porter (RP) random matrix ensemble has emerged as a minimal model for the integrability-to-chaos crossover in quantum many-body systems. Its phase diagram features a region with fractal eigenstates, exhibiting intermediate spectral and localization properties between the fully localized and fully delocalized regimes. In this work, we explore several generalizations of the RP model and determine their level statistics at the scale of the Thouless energy $E_T$, which characterizes the crossover. Using tools from free probability theory and the replica method, we compute the full counting statistics in the limit of large system size, and show that it takes a simple, universal scaling form around $E_T$, shared across all variations of the model. We validate our analytical predictions using exact numerical diagonalization of large samples, and large-deviation algorithms that resolve the full counting statistics down to probabilities as low as $10^{-40}$. We also contrast our predictions with measurements on the quantum random energy model, which is the simplest model displaying many-body localization.
Comments36+9 pages, 6+1 figures
