The quantum localization is one of the remarkable phenomena in the studies of quantum chaos and plays an important role in various contexts. Thus, an understanding of the properties of quantum localization is essential. In spite of much effort dedicated to investigating the manifestations of localization in the time-dependent systems, the features of localization in time-independent systems are still less explored, particularly in quantum systems which correspond to the classical systems with smooth Hamiltonian. In this work, we present such a study for a quantum many-body system, namely, the Dicke model. The classical counterpart of the Dicke model is given by a smooth Hamiltonian with two degrees of freedom. We examine the signatures of localization in its chaotic eigenstates. We show that the entropy localization measure, which is defined in terms of the information entropy of Husimi distribution, behaves linearly with the participation number, a measure of the degree of localization of a quantum state. We further demonstrate that the localization measure probability distribution is well described by the $\beta$ distribution. We also find that the averaged localization measure is linearly related to the level repulsion exponent, a widely used quantity to characterize the localization in chaotic eigenstates. Our findings extend the previous results in billiards to the quantum many-body system with classical counterpart described by a smooth Hamiltonian, and they indicate that the properties of localized chaotic eigenstates are universal.

中文翻译：

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Dicke模型中混沌本征态定位度量的统计性质

量子局域化是量子混沌研究中的显着现象之一，在各种情况下都起着重要的作用。因此，对量子定位特性的理解是必不可少的。尽管付出了很多努力来研究时变系统中的局域化表现，但仍很少探索时变系统中的局域化特征，特别是在量子系统中，该系统对应于具有光滑哈密顿量的经典系统。在这项工作中，我们对量子多体系统即Dicke模型进行了这样的研究。迪克模型的经典对应物是由具有两个自由度的光滑哈密顿量给出的。我们检查其混沌本征态中的本地化的签名。我们证明了熵定位测度 它是根据Husimi分布的信息熵定义的，它与参与数呈线性关系，参与数是对量子态局部化程度的度量。我们进一步证明，定位度量概率分布可以很好地描述$\beta$分配。我们还发现，平均定位度量与水平斥力指数线性相关，水平斥力指数是表征混沌本征态中定位的一种广泛使用的量。我们的发现将先前的结果扩展到了具有光滑哈密顿量描述的经典对应物的量子多体系统中，并且它们表明局部混沌本征态的性质是通用的。