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Physical Review X

Comment on “Equilibration Time Scales of Physically Relevant Observables”

Robin Heveling, Lars Knipschild, Jochen Gemmer B

Published Jun 15, 2020• 3 citations
Publisher

Abstract

Putting a generally valid upper bound on equilibration times of physically relevant observables is a much pursued endeavor. Recently, such a bound has been suggested by Garcia-Pintos et al. While the mathematical correctness of the bound as such is undisputed, its concrete calculation requires the knowledge of certain quantities, which Garcia-Pintos et al. assess by means of assumptions. We show that, e.g., in standard cases of slow, exponential equilibration, (at least) one of these assumptions is not valid. This demonstration highlights the difficulty to judge the validity of the above assumptions without further information. Such information is, in general, very hard to obtain.

DOI
10.1103/physrevx.10.028001

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Putting a generally valid upper bound on equilibration times of physically relevant observables is a much pursued endeavor. Recently, such a bound has been suggested by Garcia-Pintos et al. While the mathematical correctness of the bound as such is undisputed, its concrete calculation requires the knowledge of certain quantities, which Garcia-Pintos et al. assess by means of assumptions. We show that, e.g., in standard cases of slow, exponential equilibration, (at least) one of these assumptions is not valid. This demonstration highlights the difficulty to judge the validity of the above assumptions without further information. Such information is, in general, very hard to obtain.
One of the main questions of research on quantum many-body systems following unitary out of equilibrium dynamics is to find out how local expectation values equilibrate in time. For non-interacting models, this question is rather well understood. However, the best known bounds for general quantum systems are vastly crude, scaling unfavorable with the system size. Nevertheless, empirical and numerical evidence suggests that for generic interacting many-body systems, generic local observables, and sufficiently well-behaved states, the equilibration time does not depend strongly on the system size, but only the precision with which this occurs does. In this discussion paper, we aim at giving very simple and plausible arguments for why this happens. While our discussion does not yield rigorous results about equilibration time scales, we believe that it helps to clarify the essential underlying mechanisms, the intuition and important figure of merits behind equilibration. We then connect our arguments to common assumptions and numerical results in the field of equilibration and thermalization of closed quantum systems, such as the eigenstate thermalization hypothesis as well as rigorous results on interacting quantum many-body systems. Finally, we complement our discussions with numerical results - both in the case of examples and counter-examples of equilibrating systems.
Abstract Putting a general, physically relevant upper bound on equilibration times in closed quantum systems is a recently much pursued endeavor. In 2017 Phys. Rev. X 7 031027 García-Pintos et al suggest such a bound. We point out that the general assumptions which allow for an actual estimation of this bound are violated in cases in which Fermi’s golden rule and related open quantum system theories apply. To probe the range of applicability of Fermi’s golden rule for systems of the type addressed in the above work, we numerically solve the corresponding Schrödinger equation for some finite spin systems comprising up to 25 spins. These calculations shed light on the breakdown of standard quantum master equations in the ‘superweak’ coupling limit, which occurs for finite sized baths.
Finding the equilibration time scale is an important open question in studying the equilibration of quantum systems. There are many kinds of systems that are unable to achieve equilibration, such as Anderson insulator, many-body localization systems and some integrable systems. For those systems that can reach equilibration, it was proved that there exists a general equilibration time scale. But these upper bounds are unrealistically long, and the lower bounds are also unrealistically short. How to get an accurate and general equilibration time scale is still unclear. In this paper, we study local equilibration time scales in quantum lattice systems. With the relation between equilibration and entanglement entropy, we define a new criterion for equilibration. This criterion is based on Rényi entropy, which is simpler for calculation. Moreover, the production of Rényi entropy is highly dependent on the spreading of information, and hence the tools developed in quantum information theory can help a lot. For the upper bound of equilibration time, we use the normal criterion of the time average ofthe fluctuation of the observation. This equilibration criterion is assessed with concrete observable operator, hence it is more accurate than the criterion of Rényi entropy. But this criterion is more complicated for calculation. With an appropriate assumption about the initial state, we present a new upper bound of equilibration time. Since the results are not constrained by the Hamiltonian of the system, this bound can be applied to various situations. If we are concerned with the local equilibration, we can limit the observation to a small region. If the whole system is big enough, the local observation would always find that the rest part is staying at the canonical ensemble, so that the local equilibration time will not increase with the size of whole system. When the local region is small enough, the upper bound of equilibration time can be much shorter. The local Rényi entropy has close relation to the propagation of information. The limitation of the speed of information propagation in a quantum lattice system can be given by the Lieb-Robinson bound, with which we evaluate the production rate of local 2-Rényi entropy. With these, we give a new lower bound of equilibration time for systems whose interactions are respectively short-range, exponentially-decaying and long-range. In earlier works, the lower bound of equilibration time is equal to the Lieb-Robinson time of the local system. In our rigorous proof, however, the lower bound is related to the strength of hopping, which also decides the Lieb-Robinson velocity. This means the equilibration time is related to the Lieb-Robinson time indeed. But the strictly equal relation may not hold. These new bounds are important to understanding the process of quantum equilibration.
We investigate equilibration-on-average and effective equilibration for quantum systems evolving on infinite-dimensional Hilbert spaces under Hamiltonians with purely continuous spectrum. In this regime, the standard infinite-time dephasing picture underlying finite-dimensional equilibration theory ceases to be informative: the corresponding ergodic average becomes trivial in the relevant sense and therefore cannot serve as a physically meaningful equilibrium state. This makes equilibration intrinsically a finite-time phenomenon. Motivated by this obstruction, we introduce a framework based on finite-time ergodic averages and spectral coarse graining at finite resolution. Within this setting, we derive explicit upper bounds for equilibration-on-average of bounded observables under purely continuous-spectrum dynamics. Unlike their finite-dimensional counterparts, these bounds depend not only on the Hamiltonian and the initial state, but also on the observation time horizon and on the resolution scale of the spectral partition. We also obtain corresponding bounds for effective equilibration with respect to finite families of physically realistic POVM (Positive Operator-Valued Measure) measurements. Our results provide a quantitative formulation of equilibration for continuous-spectrum quantum systems in a regime where the usual infinite-time dephasing paradigm breaks down. Within the coarse-grained framework developed here, the resulting bounds depend explicitly on both the finite observation time and the chosen spectral resolution. An explicit example based on a free-particle Hamiltonian illustrates the mechanism and interpretation of the general bounds.