Sarkar Ayana, Kothiyal Manuja, Kumar Santosh
Department of Physics, Shiv Nadar University, Gautam Buddha Nagar, Uttar Pradesh 201314, India.
Phys Rev E. 2020 Jan;101(1-1):012216. doi: 10.1103/PhysRevE.101.012216.
The ratio of two consecutive level spacings has emerged as a very useful metric in investigating universal features exhibited by complex spectra. It does not require the knowledge of density of states and is therefore quite convenient to compute in analyzing the spectrum of a general system. The Wigner-surmise-like results for the ratio distribution are known for the invariant classes of Gaussian random matrices. However, for the crossover ensembles, which are useful in modeling systems with partially broken symmetries, corresponding results have remained unavailable so far. In this work, we derive exact results for the distribution and average of the ratio of two consecutive level spacings in the Gaussian orthogonal to unitary crossover ensemble using a 3×3 random matrix model. This crossover is useful in modeling time-reversal symmetry breaking in quantum chaotic systems. Although based on a 3×3 matrix model, our results can also be applied in the study of large spectra, provided the symmetry-breaking parameter facilitating the crossover is suitably scaled. We substantiate this claim by considering Gaussian and Laguerre crossover ensembles comprising large matrices. Moreover, we apply our result to investigate the violation of time-reversal invariance in the quantum kicked rotor system.
在研究复杂光谱所展现的普遍特征时，两个连续能级间距的比值已成为一个非常有用的度量。它不需要态密度的知识，因此在分析一般系统的光谱时计算起来相当方便。对于高斯随机矩阵的不变类，比值分布的类维格纳推测结果是已知的。然而，对于在模拟具有部分破缺对称性的系统时有用的交叉系综，到目前为止相应的结果仍然不可得。在这项工作中，我们使用一个3×3随机矩阵模型，推导出高斯正交到酉交叉系综中两个连续能级间距比值的分布和平均值的精确结果。这种交叉在模拟量子混沌系统中的时间反演对称性破缺时很有用。尽管基于一个3×3矩阵模型，但只要适当缩放促进交叉的对称性破缺参数，我们的结果也可应用于大光谱的研究。我们通过考虑包含大矩阵的高斯和拉盖尔交叉系综来证实这一说法。此外，我们应用我们的结果来研究量子踢转子系统中时间反演不变性的破坏。
