Onset of chaos and relaxation in isolated systems of interacting spins: Energy shell approach

We study the onset of chaos and statistical relaxation in two isolated dynamical quantum systems of interacting\r\nspins 1/2, one of which is integrable and the other chaotic. Our approach to identifying the emergence of chaos is\r\nbased on the level of delocalization of the eigenstates with respect to the energy shell, the latter being determined by the interaction strength between particles or quasiparticles. We also discuss how the onset of chaos may be anticipated by a careful analysis of the Hamiltonian matrices, even before diagonalization. We find that despite\r\ndifferences between the two models, their relaxation processes following a quench are very similar and can be\r\ndescribed analytically with a theory previously developed for systems with two-body random interactions. Our\r\nresults imply that global features of statistical relaxation depend on the degree of spread of the eigenstates within the energy shell and may happen to both integrable and nonintegrable systems.