随机矩阵本征态的Hilbert空间几何
Hilbert-space geometry of random-matrix eigenstates
摘要：
量子态的多参数族的几何结构在许多背景下都很重要，包括绝热或非绝热量子动力学、量子猝灭以及量子临界点的表征。在这里，我们讨论了参数依赖的随机矩阵系综的本征态的希尔伯特空间几何，导出了高斯酉系综的量子几何张量的全概率分布。我们的分析结果给出了Fubini-Study度量和Berry曲率的精确联合分布函数。我们讨论了Levy稳定分布的关系，并将我们的结果与随机矩阵集合以及随机磁场中的电子的数值模拟进行了比较。
英文摘要：
The geometry of multi-parameter families of quantum states is important in numerous contexts, including adiabatic or nonadiabatic quantum dynamics, quantum quenches, and the characterization of quantum critical points. Here, we discuss the Hilbert-space geometry of eigenstates of parameter-dependent random-matrix ensembles, deriving the full probability distribution of the quantum geometric tensor for the Gaussian Unitary Ensemble. Our analytical results give the exact joint distribution function of the Fubini-Study metric and the Berry curvature. We discuss relations to Levy stable distributions and compare our results to numerical simulations of random-matrix ensembles as well as electrons in a random magnetic field.
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