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1 Sep 2015 Quantum Many-Body Systems out of Equilibrium View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by UPCommons. Portal del coneixement obert de la UPC Quantum many-body systems out of equilibrium J. Eisert, M. Friesdorf, and C. Gogolin Dahlem Center for Complex Quantum Systems, Freie Universitat¨ Berlin, 14195 Berlin, Germany Closed quantum many-body systems out of equilibrium pose several long-standing problems in physics. Recent years have seen a tremendous progress in approaching these questions, not least due to experiments with cold atoms and trapped ions in instances of quantum simulations. This article provides an overview on the progress in understanding dynamical equilibration and thermalisation of closed quantum many-body systems out of equilibrium due to quenches, ramps and periodic driving. It also addresses topics such as the eigenstate thermalisation hy- pothesis, typicality, transport, many-body localisation, universality near phase transitions, and prospects for quantum simulations. How do closed quantum many-body systems out of equi- such as the Ising model, as well as bosonic or fermionic lat- librium eventually equilibrate? How and in precisely what tice systems, prominently Fermi- and Bose-Hubbard models. way can such quantumsystems with many degrees of freedom In the latter case, the Hamiltonian takes the form thermalise? It was established early on how to capture equilib- † † rium states corresponding to some temperature in the frame- H = J (bj bk + bkbj )+ U nj (nj − 1), (2) work of quantum statistical mechanics, but it seems much less hXj,ki Xj clear how such states are eventually reached via the local dy- † namics following microscopic laws. This dichotomy gives with nj = bj bj , bj denoting the bosonic annihilation operator rise to some tension between a microscopic description and at site j, the parameters U,J being real and the sum hj, ki one in terms of the familiar ensembles of statistical mechan- running over nearest neighbours only. Common variants of ics. It also leads to the question of how the actual dynamics of this are models with other short range interactions, or long quantum phase transitions can be grasped. Asking questions range interactions with a strength decaying polynomially in of this type has a long tradition, but many problems have re- the distance. mained largely unresolved. It was only relatively recently that they moved back into the focus of attention. This development Equilibration after quenches is driven by the availability of experiments that allow to probe such questions under controlled conditions, but also by new developments in theoretical physics. The present review aims Investigating the fundamental connection between thermo- dynamics and the dynamics of closed quantum systems has a at providing a first impression of important progress in this 1 rapidly developing field of research, it emphasises promising long tradition but has also been a key topic in recent years. A very clean setting for non-equilibrium dynamics is the one perspectives, and collects a number of key references. Given 2–9,11–16 its brevity, it can by no means be faithful to all developments emerging from a sudden global quench : Initially, the or cover the whole literature of the field. system is in a state ρ0, which could be the ground state of a local Hamiltonian. Then one quickly alters the system’s pa- arXiv:1408.5148v2 [quant-ph] 1 Sep 2015 rameters globally and considers the many-body unitary time Local quantum many-body systems evolution under some local Hamiltonian H. Of specific inter- est are expectation values of observables A at later times The physical systems considered here are local quantum −itH itH many-body systems with finite-range interactions, ubiquitous hA(t)i = tr(e ρ0 e A). (3) in condensed-matter physics. They can be described by some A main question is to what extent and for what times the sit- interaction graph, with physical constituents at the vertices. uation can be described by a suitable equilibrium ensemble. The system is governed by a Hamiltonian that can be written Notably, the dynamics is time reversal invariant and for finite as systems recurrent, so a priori it seems far from clear how and in what sense equilibrium can be reached dynamically. H = hj . (1) Xj A first important insight is that such systems generically indeed relax and equilibrate in the following sense: Even Most commonly considered are nearest-neighbour interac- though the dynamics is entirely unitary, following transient tions, where each hj acts non-trivially on finitely many sites. non-equilibrium dynamics, it is expectation values hA(t)i of Paradigmatic examples of this type are spin lattice systems, many observables that equilibrate. This is specifically true for 2 local observables which are supported only on a small num- esting research topic in its own right. ber of sites. The apparent long time equilibrium state has to be equal to the time average Thermalisation T 1 −itH itH ω = lim dt e ρ0 e . (4) T →∞ T Z0 The success of thermodynamics in describing large scale systems indicates that often the equilibrium values can be de- This state is the maximum entropy state, holding all constants scribed by few parameters such as the global temperature and 17 of motion fixed . In case of non-degenerate eigenvalues of particle number. We use the term thermalisation to refer to the the Hamiltonian, this ensemble is also called the diagonal en- equilibration towards a state ω that is in a suitable sense close 18 semble . This feature is reminiscent of a dynamical Jaynes’ to being indistinguishable from a thermal equilibrium state principle: A many-body system is pushed out of equilibrium proportional to e−βH for some inverse temperature β > 0. and follows unitary dynamics. Yet, for most times, the sys- There are several mechanisms that can lead to thermalisation tems appears as if it had equilibrated to a maximum entropy in this sense: The first is based on the eigenstate thermalisa- state. tion hypothesis (ETH)9,37–40. It conjectures that sufficiently The general expectation that many-body systems equili- complex quantum systems have eigenstates that — for phys- brate can be made rigorous in a number of senses: For some ically relevant observables such as local ones — are practi- free models, specifically for the case of the integrable non- cally indistinguishable from thermal states with the same av- interacting limit of the Bose-Hubbard model, equilibration is erage energy. The ETH and its breakdown in integrable sys- proven to be true in the strong sense of equilibration during tems have recently been investigated extensively, mostly with intervals: Here, one can show for local observables A that numerical methods. Supporting evidence for it has been col- |hA(t)i− tr(ωA)| is arbitrarily small after a known initial re- lected in a plethora of models9,12,41–43. laxation time, and remains so for a time that grows linearly with the system size6,19. One merely has to assume that the Without invoking the ETH one can still rigorously show thermalisation in weakly interacting systems under stronger otherwise arbitrary initial state has suitable polynomially de- 44 caying correlations. At the heart of such a rigorous argument conditions on the initial state . In addition, dynamical ther- malisation of local expectation values to those in the global are Lieb-Robinson bounds as well as non-commutativecentral 45 limit theorems. thermal state has been shown under stronger conditions, most notably translation invariance and the existence of a If no locality structure is available or is made use of, one unique Gibbs state. The notion of relative thermalisation46 can still show equilibration on average for all Hamiltoni- focuses on the decoupling of a subsystem from a reference ans that have non-degenerate energy gaps: One can bound system. So far, there is no compelling proof that globally the time average Et∈[0,∞)|hA(t)i − tr(ωA)| by a quantity quenched non-integrable systems dynamically thermalise. that is usually exponentially small in the system size20,21. This means that out of equilibrium quantum systems ap- These dynamical approaches are complemented by works 47,48 pear relaxed for overwhelmingly most times, even if by that based on typicality arguments . Here, rather than following statement alone no information on time scales can be de- the dynamical evolution of a system, one tries to justify the duced. More refined bounds have been derived, also pro- applicability of ensembles by showing that most states drawn viding some information about the relevant time scales of randomly according to some measure have the same physical equilibration22,23, but they are far off from those observed in propertiesas some appropriate ensemble such as the canonical numerical simulations4,7,12,16 and experiments14,24,25. These one. findings are also related to further efforts of describing the 49 26,27 If a quantum system is integrable in the sense that it has size of the time fluctuations after relaxation . local conserved quantities, one should not expect the system By no means are sudden global quenches the only rele- to thermalise: These constants of motion prohibit full ther- vant setting of quantum many-body systems out of equilib- malisation to the canonical ensemble. One can still expect rium. Similarly important are scenarios of local quenches28,29, the system to equilibrate to the maximum entropy state given where not the entire system is uniformly modified, but rather these locally conserved quantities15,17, a so called generalized the system is locally suddenly driven out of equilibrium Gibbs ensemble6,18. Effective thermalisation has been put into by a change limited to a bounded number of lattice sites. context with unitary quantum dynamics under conditions of Ramps12,30–33, during which the Hamiltonian is changed ac- classical chaos50.
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