Sarkar Ayana, Dheer Ashutosh, Kumar Santosh
Department of Physics, Shiv Nadar Institution of Eminence (SNIoE), Gautam Buddha Nagar, Uttar Pradesh 201314, India.
Chaos. 2024 Mar 1;34(3). doi: 10.1063/5.0180902.
Multifractal analysis is a powerful approach for characterizing ergodic or localized nature of eigenstates in complex quantum systems. In this context, the eigenvectors of random matrices belonging to invariant ensembles naturally serve as models for ergodic states. However, it has been found that the finite-size versions of multifractal dimensions for these eigenvectors converge to unity logarithmically slowly with increasing system size N. In fact, this strong finite-size effect is capable of distinguishing the ergodicity behavior of orthogonal and unitary invariant classes. Motivated by this observation, in this work, we provide semi-analytical expressions for the ensemble-averaged multifractal dimensions associated with eigenvectors in the orthogonal-to-unitary crossover ensemble. Additionally, we explore shifted and scaled variants of multifractal dimensions, which, in contrast to the multifractal dimensions themselves, yield distinct values in the orthogonal and unitary limits as N→∞ and, therefore, may serve as a convenient measure for studying the crossover. We substantiate our results using Monte Carlo simulations of the underlying crossover random matrix model. We then apply our results to analyze the multifractal dimensions in a quantum kicked rotor, a Sinai billiard system, and a correlated spin-chain model in a random field. The orthogonal-to-unitary crossover in these systems is realized by tuning relevant system parameters, and we find that in the crossover regime, the observed finite-dimension multifractal dimensions can be captured very well with our results.
多重分形分析是一种用于刻画复杂量子系统中本征态遍历性或局域性的有力方法。在此背景下，属于不变系综的随机矩阵的本征向量自然地充当遍历态的模型。然而，已经发现这些本征向量的多重分形维数的有限尺寸版本随着系统尺寸(N)的增加以对数方式缓慢收敛到(1)。事实上，这种强烈的有限尺寸效应能够区分正交和酉不变类的遍历性行为。受此观察结果的启发，在这项工作中，我们给出了与正交到酉交叉系综中的本征向量相关的系综平均多重分形维数的半解析表达式。此外，我们探索了多重分形维数的平移和缩放变体，与多重分形维数本身不同，当(N→∞)时，它们在正交和酉极限中产生不同的值，因此可以作为研究交叉的一种方便度量。我们使用基础交叉随机矩阵模型的蒙特卡罗模拟来证实我们的结果。然后我们将结果应用于分析量子受踢转子、辛钦台球系统和随机场中的相关自旋链模型中的多重分形维数。这些系统中的正交到酉交叉是通过调整相关系统参数实现的，并且我们发现在交叉区域，我们的结果能够很好地捕捉到观测到的有限维多重分形维数。
