Possible strategies to describe analytically the dynamics of many-body quantum systems out of equilibrium include the use of solvable models and of full random matrices. None of the two approaches represent actual realistic systems, but they serve as references for the studies of these ones. We take the second path and obtain analytical expressions for the survival probability, density imbalance, and out-of-time-ordered correlator. Using these findings, we then propose an approximate expression that matches very well numerical results for the evolution of realistic finite quantum systems that are strongly chaotic and quenched far from equilibrium. In the case of the survival probability, the expression proposed covers all different time scales, from the moment the system is taken out of equilibrium to the moment it reaches a new equilibrium. The realistic systems considered are described by one-dimensional spin-1/2 models.
It is known that the Sachdev-Ye-Kitaev (SYK) model is exactly solvable at leading order in
1/q; specifically, the instantaneously thermal time block after the quench is known in closed form. Thus far the remaining (inherently out of equilibrium) blocks have only been numerically accessible. Here we provide their analytical solutions for a quench between non-commuting
q/2-body SYK Hamiltonians. We obtain the Green's functions in closed form for every time block. We extract a simple relation between the pre- and post-quench energies
ε1∝ε0 leading to an exact temperature update rule. Via a Cauchy-Schwarz inequality, we show that heating is inevitable for the closed system; hence, the second law emerges geometrically. The solution is used in a companion paper to address what ``instantaneous thermalization'' means at finite
q.
I will give an overview of recent work on minimal models for quantum chaos in spatially extended many-body quantum systems, describing simple, solvable models. The detailed study of generic quantum systems - ones without exact or approximate conservation laws - goes back at least as far as investigations of highly-excited states in nuclei in the 1950s. It saw revivals in the 1980s with work on systems such as quantum billiards, that have only a few degrees of freedom, and also in the context of mesoscopic conductors. It is attracting renewed current interest with a focus on many-body systems that are spatially extended. Part of the motivation for this comes from experiments on cold atom systems, and part comes from development on the theoretical understanding of eigenstates of many-body systems. The study of dynamics in spatially many-body systems introduces new questions that do not arise in finite systems such as quantum billiards, or in extended single-particle systems, such as mesoscopic conductors. The questions concern the dynamics of quantum information and the approach to equilibrium. I will discuss how these can be answered in the context of minimal models. References: A. Chan, A. De Luca, and J. T. Chalker, Phys. Rev. Lett. 121, 060601 (2018) and Phys. Rev. X 8, 041019 (2018)