随机厄米矩阵诱导的量子系统中混沌演化的表征
Characterization of Chaotic Evolution in Quantum Systems Induced by Random Hermitian Matrices
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中文总结 AI 辅助
本文将半经典李雅普诺夫指数形式主义应用于五种随机矩阵系综，推导相关物理量并对比数值与理论结果，以表征随机厄米矩阵诱导的量子系统混沌演化。
中文摘要 AI 辅助
在近期一篇论文中，与有限维厄米矩阵表示的量子哈密顿量相关的半经典李雅普诺夫指数被定义并奠定了数学基础。该李雅普诺夫指数表征系统向遍历态演化的早期阶段，而晚期阶段则由对应马尔可夫矩阵的谱隙表征。本文将该形式主义应用于五种随机矩阵系综，对每个系综推导平均李雅普诺夫指数、其方差及谱隙作为能量的函数，并给出对应的热平均，将大量数值数据与理论预测进行对比。
英文摘要
In a recent paper, a semiclassical Lyapunov exponent associated with a quantum Hamiltonian represented by a finite-dimensional Hermitian matrix was defined and placed on a mathematical foundation. The Lyapunov exponent characterizes the early stages of the evolution toward the ergodic state, while the late stages are characterized by the spectral gap of the corresponding Markov matrix. Here, we apply this formalism to five random-matrix ensembles. For each ensemble, we derive the mean Lyapunov exponent, its variance, and the spectral gap as functions of energy. We also present the corresponding thermal averages. Extensive numerical data are compared with the theoretical predictions.
