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arXiv:1610.03770 · Journal of Statistical Mechanics Theory and Experiment

N -scaling of timescales in long-range N -body quantum systems

Michael Kästner

Published Jan 1, 2017• 10 citations
arXivHTMLPDF

Abstract

Abstract Long-range interacting many-body systems exhibit a number of peculiar and intriguing properties. One of those is the scaling of relaxation times with the number N of particles in a system. In this paper I give a survey of results on long-range quantum spin models that illustrate this scaling behaviour, and provide indications for its common occurrence by making use of Lieb–Robinson bounds. I argue that these findings may help in understanding the extraordinarily short equilibration timescales predicted by typicality techniques.

DOI
10.1088/1742-5468/aa5119

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Abstract Long-range interacting many-body systems exhibit a number of peculiar and intriguing properties. One of those is the scaling of relaxation times with the number N of particles in a system. In this paper I give a survey of results on long-range quantum spin models that illustrate this scaling behaviour, and provide indications for its common occurrence by making use of Lieb–Robinson bounds. I argue that these findings may help in understanding the extraordinarily short equilibration timescales predicted by typicality techniques.
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.
Abstract We state and prove four types of Lieb–Robinson bounds valid for many-body open quantum systems with power law decaying interactions undergoing out of equilibrium dynamics. We also provide an introductory and self-contained discussion of the setting and tools necessary to prove these results. The results found here apply to physical systems in which both long-ranged interactions and dissipation are present, as commonly encountered in certain quantum simulators, such as Rydberg systems or Coulomb crystals formed by ions.
We state and prove four types of Lieb–Robinson bounds valid for many-body open quantum systems with power law decaying interactions undergoing out of equilibrium dynamics. We also provide an introductory and self-contained discussion of the setting and tools necessary to prove these results. The results found here apply to physical systems in which both long-ranged interactions and dissipation are present, as commonly encountered in certain quantum simulators, such as Rydberg systems or Coulomb crystals formed by ions.
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.
In this paper, we prove a family of Lieb-Robinson bounds for discrete spin systems with long-range interactions. Our results apply for arbitrary kkk-body interactions, so long as they decay with a power law greater than kdkdkd, where ddd is the dimension of the system. More precisely, we require that the sum of the norm of terms with diameter greater than or equal to RRR, acting on any one site, decays as a power law $1/{R}^{\ensuremath{\alpha}}$, with $\ensuremath{\alpha}>d$. These bounds allow us to prove that, at any fixed time, the spatial decay of a time evolved operator follows arbitrarily closely to $1/{r}^{\ensuremath{\alpha}}$. Moreover, we introduce an alternative light cone definition for power-law interacting quantum systems which captures the region of the system where changing the Hamiltonian can affect the evolution of a local operator. In short-range interacting systems, this light cone agrees with the conventional definition. However, in long-range interacting systems, our definition yields a stricter light cone, which is more relevant in most physical contexts.