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

Abstract

We study the onset of chaos and statistical relaxation in two isolated dynamical quantum systems of interacting spins 1/2, one of which is integrable and the other chaotic. Our approach to identifying the emergence of chaos is based 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 differences between the two models, their relaxation processes following a quench are very similar and can be described analytically with a theory previously developed for systems with two-body random interactions. Our results 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.

Identifying the closeness of eigenstates in quantum many-body systems

“…The reduction of fluctuation was also found in ref. [9] by using a similar measure as discussed above. But except for this result, the closeness are measured differently.…”

Effects of the interplay between initial state and Hamiltonian on the thermalization of isolated quantum many-body systems

“…Two of them involve only integrable Hamiltonians (λ = 0), ∆ being the parameter changed: Notice that for the quenches above, the perturbation is strong, as shown in Refs. [33,34]. As the perturbation increases from zero, the shape of the initial state (that is, the distribution of the components |C n α | 2 in E α ) expands from a delta function to a Breit-Wigner (Lorentzian) form and eventually becomes a Gaussian function as determined by the energy shell.…”

“…(7), the width of the shell is related to the number of states that are directly coupled with the initial state, that is, it quantifies the level of connectivity of the initial state [33,34]. The lowest connectivity is seen for the quench XXZ → XX, which also fluctuates significantly.…”

“…When dealing with realistic systems of few-body interactions, a completely chaotic state is defined as a state that fills the energy shell [31][32][33][34].…”

“…Thermalization is assured for any initial state that fills the energy shell ergodically. In this case its coefficients become uncorrelated random variables following a Gaussian distribution [31][32][33][34]. Any eigenstate from a full random matrix projected on the energy eigenbasis of a delocalized system falls in this category.…”