Fig. 1: Typical late-time behavior of the spectral form factor K(t), where t_{H} is the Heisenberg time. (Source: J. Wang) |
Beginning with the pioneering work by Wigner, Random Matrix Theory (RMT) has been a fruitful diagnostic of quantum chaos. Wigner originally proposed RMT as a statistical description of the spectra of large atomic systems, whose interactions are so fascinatingly complex that they can essentially be treated as ensembles of random Hamiltonians drawn from a class respecting the symmetries of the problem. Surprisingly, such an approach works even for single-particle systems with chaotic classical analogues. As such, the agenda of RMT in quantum chaos has been to classify Hamiltonian ensembles into universality classes associated with predetermined random matrix ensembles based on certain wisely-chosen metrics, as the system size tends to infinity.
In this paper, one particular metric known as the spectral form factor is reviewed and applied as a probe in examples of chaotic Floquet systems, with and without classical counterparts. The typical behavior of the spectral form factor K(t) at times larger than a time-scale known as the Thouless time, after which universality is expected to set in, is depicted in Fig. 1. For a Floquet system with period τ and Hilbert space dimension |N|, the spectral form factor is defined as
where U is the propagator over a single period τ. From random matrix theory, the spectral form factors of the Circular Orthogonal Ensemble (COE) and Circular Unitary Ensemble (CUE) expected to be the universality classes of chaotic Floquet systems with and without time-reversal symmetry are given by as: [1]
K_{COE}(t) | = | 2t - t ln(1 + 2t) |
= | 2t - 2t^{2} +O(t^{3}) | |
K_{CUE}(t) | = | t |
for times t smaller than the Heisenberg time. In the full paper, we review how Floquet systems with chaotic classical analogs indeed fall into these universality classes by verifying their spectral form factors up to first order in t. In this process, the Hannay-Ozorido de Alemida sum rule for periodic paths will be invoked. [2] We also demonstrate this first-order agreement for a particular system with periodic Hamiltonian
where H_{0} is a many-body-localized system composed of l-bits and H_{1} is an on-site driving force. This work is a review of the original paper by Kos et al. [3]
© Jinhui Wang. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] M. L. Mehta, Random Matrices and the Statistical Theory of Spectra, 2nd Ed. (Academic, New York, 1991).
[2] J. H. Hannay, A. M. Ozorio De Almeida, "Periodic Orbits and a Correlation Function for the Semiclassical Density of States," J. Phys. A: Math. Gen. 17, 3429 (1984).
[3] P. Kos, M. Ljubotina, and T. Prosen, "Many-Body Quantum Chaos: Analytic Connection to Random Matrix Theory," Phys. Rev. X 8, 021062 (2018).