论文标题
Loschmidt光谱形式因素
The Loschmidt Spectral Form Factor
论文作者
论文摘要
光谱形式（SFF）测量了哈密顿量状态密度的波动。我们考虑称为loschmidt光谱符号的SFF的概括，$ \ textrm {tr} [e^{ih_1t}] \ textrm {tr} [e^{ - ih_2t}] $，对于$ h_1-h_2 $ small。如果SFF的合奏平均值是从合奏中绘制的单个汉密尔顿的密度波动的差异，则平均的loschmidt SFF是从相关集合中抽出的两个汉密尔顿人的协方差。该对象是在量子混乱和随机矩阵文献中研究的参数相关的时间域版本。我们分析表明，对于复杂的速率$λ$，平均的Loschmidt SFF与$ e^{iλt} t $成正比，具有积极的想象零件，以定量的方式表明，频谱的长期细节对扰动的长期细节比较敏感。在许多情况下，我们计算$λ$，包括随机矩阵理论，具有单个局部缺陷的理论和流体动力学理论。
The Spectral Form Factor (SFF) measures the fluctuations in the density of states of a Hamiltonian. We consider a generalization of the SFF called the Loschmidt Spectral Form Factor, $\textrm{tr}[e^{iH_1T}]\textrm{tr} [e^{-iH_2T}]$, for $H_1-H_2$ small. If the ensemble average of the SFF is the variance of the density fluctuations for a single Hamiltonian drawn from the ensemble, the averaged Loschmidt SFF is the covariance for two Hamiltonians drawn from a correlated ensemble. This object is a time-domain version of the parametric correlations studied in the quantum chaos and random matrix literatures. We show analytically that the averaged Loschmidt SFF is proportional to $e^{iλT}T$ for a complex rate $λ$ with a positive imaginary part, showing in a quantitative way that the long-time details of the spectrum are exponentially more sensitive to perturbations than the short-time properties. We calculate $λ$ in a number of cases, including random matrix theory, theories with a single localized defect, and hydrodynamic theories.
