Stability of quantum dynamics under constant Hamiltonian perturbations

Abstract

Concepts like "typicality" and the "eigenstate thermalization hypothesis" aim at explaining the apparent equilibration of quantum systems, possibly after a very long time. However, these concepts are not concerned with the specific way in which this equilibrium is approached. Our point of departure is the (evident) observation that some forms of the approach to equilibrium, such as, e.g., exponential decay of observables, are much more common then others. We suggest to trace this dominance of certain decay dynamics back to a larger stability with respect to generic Hamiltonian perturbations. A numerical study of a number of examples in which both, the unperturbed Hamiltonians as well as the perturbations are modelled by partially random matrices is presented. We furthermore develop a simple heuristic, mathematical scheme that describes the result of the numerical investigations remarkably well. According to those investigations the exponential decay indeed appears to be most stable. Dynamics that are in a certain sense at odds with the arrow of time are found to be very unstable.

Typical relaxation of perturbed quantum many-body systems

“…In particular, a purely exponential decay arises when A ρ0(t) in ( 16) is time-independent, for instance, since the observable A is a constant of motion, or ρ 0 (t) is a steady state of the unperturbed system. Phenomenologically, such "exponential laws" are extremely common, and our theory substantially extends previous attempts to explain why this is so even in isolated systems, see [22,27,[47][48][49] and references therein.…”

“…In any case, it already affirms the general structure (33a) with d n1n2 µ1µ2 and f n1n2 µ1µ2 to be determined. To compute the correlator (76), we proceed as for the second moment and aim at extracting it from the product of average resolvents similarly to (49). As we are dealing with a product of four overlap matrices now, we have to consider products of two resolvents.…”

Magnetization and energy dynamics in spin ladders: Evidence of diffusion in time, frequency, position, and momentum

“…Following an approach introduced in Ref. [85], the dynamics p q (t) ∝ J 0 (2t) is interpreted as being generated by an integro-differential equation comprising a memory kernel K(t), 21) and (23)] with damping γ = 0.6. The XY data is multiplied by a factor of 2 in order to account for the two legs of the ladder.…”

Nontrivial damping of quantum many-body dynamics

“…A particularly intriguing and omnipresent question in physics is how the dynamics of a given quantum system is affected by the presence of a perturbation [12,[22][23][24][25][26], i.e., scenarios where the Hamiltonian H of the full system can be written as…”