View metadata,citationandsimilarpapersatcore.ac.uk arXiv:1408.5148v2 [quant-ph] 1 Sep 2015 oa unu aybd systems many-body quantum Local address simulations. also It localisation many-body driving. transport, typicality, periodic pothesis, and ramps quenches, to due thermalisatio simulations. and quantum equilibration dynamical of understanding q instances in these ions approaching trapped in and progress atoms tremendous se a pose seen equilibrium have of years out systems many-body quantum Closed Universit Freie Systems, Quantum Complex for Center Dahlem Gogolin C. and Friesdorf, M. Eisert, equilibrium J. of out systems many-body Quantum t rvt,i a yn en efihu oaldevelopment field. the all of to literature faithful whole be the means cover no Give or by references. can key it brevity, of its number this a promisi in collects emphasises and it progress perspectives, research, important of of field impression developing rapidly first a ai new review providing present by at The also physics. but theoretical in conditions, developments controlled pr to under allow that questions experiments such of availability developmen the This by driven attention. is of focus re- the have into recently back problems relatively moved many only they was but It tradition, unresolved. long largely a questions mained has Asking type o grasped. this dynamics be of actual the can how transitions of mechan phase question statistical the quantum to of leads ensembles also and familiar It description the ics. microscopic of a terms gives in between dichotomy one tension This some dy to local laws. rise the microscopic via reached following eventually namics are les states much such seems how it frame- clear but the mechanics, in statistical quantum temperature of some work to equil corresponding capture states to how rium on early what established freedom was of precisely It degrees many thermalise? in with systems and quantum such How can way equilibrate? eventually librium h ytmi oendb aitna htcnb written be can that Hamiltonian verti a as the by governed at is constituents system The physical some with by described graph, be interaction can They physics. condensed-matter in systems many-body otcmol osdrdaenaetnihorinterac- nearest-neighbour each are where tions, considered commonly Most aaimtceape fti yeaesi atc systems lattice spin are type this of examples Paradigmatic h hsclssescniee eeare here considered systems physical The equi- of out systems many-body quantum closed do How h j ihfiierneitrcin,ubiquitous interactions, finite-range with csnntiilyo ntl aysites. many finitely on non-trivially acts H = X j h j . oa quantum local ces. that nvraiyna hs rniin,adpopcsfrqu for prospects and transitions, phase near universality , obe ms (1) ib- ng n s s f - - , t fcoe unu aybd ytm u fequilibrium of out systems many-body quantum closed of n nedrlxadeulbaei h olwn es:Even values transie expectation following is sense: unitary, it following entirely dynamics, is non-equilibrium the dynamics in the equilibrate though and relax indeed dynamically. and reached how be clear can from equilibrium far sense seems what it in fini priori for a and so ensemble. invariant recurrent, reversal equilibrium systems sit- time suitable the is times a dynamics what the by Notably, for described and be extent can what uation to is question main A yaisadtednmc fcoe unu ytm a a has systems quantum tradition closed long of dynamics the and dynamics s r xetto auso observables of values expectation are est eycenstigfrnneulbimdnmc steone the sudden is a dynamics from non-equilibrium emerging for setting clean very A aeesgoal n osdr h aybd ntr time Hamiltonian unitary local many-body some pa- under system’s the evolution the considers alters and quickly globally one rameters Then Hamiltonian. state local a in is system quenches after Equilibration ayobservables many ag neatoswt teghdcyn oyoilyi long polynomially or decaying interactions, strength distance. the range a short with of other interactions variants range with Common models are only. this neighbours nearest over running ntelte ae h aitna ae h form the takes Hamiltonian lat- the fermionic case, models. latter or Bose-Hubbard the bosonic In and as Fermi- well prominently as systems, model, tice Ising the as such with tsite at rtipratisgti htsc ytm generically systems such that is insight important first A netgtn h udmna oncinbtenthermo- between connection fundamental the Investigating tBri,115Bri,Germany Berlin, 14195 Berlin, at hsatcepoie noeve ntepors in progress the on overview an provides article This ¨ stpc uha h iesaetemlsto hy- thermalisation eigenstate the as such topics es n ea ogsadn rbesi hsc.Recent physics. in problems long-standing veral j etos o es u oeprmnswt cold with experiments to due least not uestions, j H h parameters the , = = b j † J b 1 j h X , u a lobe e oi nrcn years. recent in topic key a been also has but j,k h b A j i hteulbae hsi pcfial refor true specifically is This equilibrate. that ( ( eoigtebsncanhlto operator annihilation bosonic the denoting b t ρ ) j † 0 i b hc ol etegon tt fa of state ground the be could which , k = + lblquench global ,J U, tr provided byUPCommons.PortaldelconeixementobertdelaUPC b ( k † e b − j en eladtesum the and real being i + ) tH ρ U 0 e X i tH j 2 – H A 9 A n , 11 fseicinter- specific Of . ) tltrtimes later at j . – ( 16 n j ntal,the Initially, : − brought toyouby 1) h A , antum ( t h ) ,k j, i (3) (2) of nt te n i CORE 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. cording to some schedule, are of interest in studies of the dy- namics of quantum phase transitions. The response of a sys- Several quantum many-body systems exhibit some instance tem to a sudden alteration of its geometry can be studied by of pre-thermalisation: This term refers to any apparent equi- geometric quenches5,34,35. Time-dependent periodic driving is libration to a meta-stable state on a short time scale, before discussed in a separate section. In this brief review, closed — on a long time scale — the system relaxes to a state in- many-body systems are in the focus of attention. It should be distinguishable from a genuine thermal state. Examples are 51,52 clear that specifically pump-probe type settings36 generate an almost integrable systems and continuous systems of cou- 25 out of equilibrium situation that can be used as a diagnostic pled Bose-Einstein condensates . tool in the condensed-matter context, giving rise to an inter- 3

“Light cones” and entanglement dynamics optical lattices one commonlystudies the transport of particles starting from an initially trapped situation72,73, or of quasi- An important stepping stone for a better understanding of particles57,58. Two of the key aims are to distinguish between the non-equilibrium dynamics of local Hamiltonian models is diffusive and ballistic transport and to understand the limita- the insight that Lieb-Robinson bounds53–55 limit the speed of tions of linear response theory71. While simple, integrable information propagation in such systems. There are several models can be solved exactly and thus provide an excellent ways of stating such bounds: One is to say that for any two benchmark, our understanding of transport in realistic setups observables A and B heavily relies on numerical tools. Tools explicitly storing the full wave function in the memory, such as exact diagonalisa- k[A(t),B]k≤ c kAkkBk min{|A|, |B|} e−µ (d(A,B)−v |t|), tion, are limited to comparably small systems. Tensor network (5) tools such t-DMRG, TEBD, and variants thereof63,65 allow to where d(A, B) is the distance between the support of the ob- parametrize the state more efficiently and are accurate up to servables, |A| and |B| the size of their supports, v ≥ 0 takes machine precision as long as entanglement entropies are suf- the role of a group velocity, and c,µ > 0 are constants. That ficiently small74. Thus, for short and intermediate times these is to say, information propagation outside a “light-cone” is tools allow for an investigation of transport and equilibration exponentially suppressed. It immediately follows that corre- in systems with several hundred sites14,16,71,73. lation functions can significantly grow in time only inside the 55 light cone . While these bounds are not necessarily tight, the Absence of thermalisation and many-body localisation picture that the local dynamics leads to excitations traveling through the system with velocity v offers a lot of explanatory Crucial for transport properties of materials is the influence 3,6,7,19,24,56,57 potential for non-equilibrium dynamics . Such of irregularities such as defects. The impact of random poten- light-cone dynamics has been experimentally observed in op- tials on particle mobility has thus been the subject of inten- tical lattice systems using novel tools that allow for single-site sive study over the last century, culminating in the famous re- 58 25 addressing and in continuous systems of cold atoms . sult of Anderson, showing full localisation both of the eigen- The light-cone-like propagationof information has implica- functions as well as of the dynamics75 for single particle dis- tions on entanglement dynamics. Many-bodystates are said to ordered models. In the case of non-interacting fermions in satisfy an area law for the entanglement entropy, if for a sub- one dimension or equivalently, via the Jordan-Wigner trans- set of sites the latter scales only like the boundary“area” ofthe formation, for free spin chains, Anderson’s result directly subset rather than its “volume”59,60. It has been shown that if gives strong localisation of all fermionic modes. This leads initially such a law holds, this will remain to be true for longer to the entanglement entropy remaining upper bounded from times55,61. Yet, at the same time, the pre-factor of this area above76 and a vanishing Lieb-Robinson velocity77. These sys- law is expected to grow exponentially62. This is the ultimate tems thus fail to serve as their own heat bath and transport reason why numerical tensor network methods63,64, such as of any kind is completely suppressed. For interacting many- the density-matrix renormalisation group method65, can sim- body systems, the effects of disorder are still much less clear, ulate out of equilibrium dynamics efficiently for short times despite great efforts78. Interestingly, both the support of time on a classical computer, while long times are not accessible. evolved local observables as well as the entanglement entropy This is a feature shared by other powerful numerical meth- are found to grow logarithmically in the interacting case and ods as well, such as Monte-Carlo66 and dynamical mean field are thus unbounded76, in stark contrast to the non-interacting theory (DMFT)67. This is in line with complexity-theoretic case. Dynamical aspects of many-body localisation are ex- arguments that provide evidence that generic quantum long- pected to go along with localisation of the eigenstates. In time dynamics cannot be efficiently classically simulated68. several recent studies localisation effects due to disorder have If interactions are not strictly local, modified Lieb-Robinson been associated with a lack of entanglement in many energy bounds still hold53, giving rise to a rich phenomenologywhich eigenstates17,79,80 or alternatively only in those below a so has been experimentally explored in systems of trapped ions called mobility edge77,78. This could lead to a phase transi- exhibiting long-range order57,69. tion between a disordered insulator with exactly zero conduc- tivity and a conducting phase78. The localisation of eigen- Transport states leads to a clear violation of the ETH and connects to an absence of thermalisation effect for localising many-body Understanding electronic transport is one of the main moti- systems and the existence of local constants of motion in these 17,79 vations that started the field of condensed matter physics and models . While these signatures of many-body localisation gave rise to a large body of work trying to accurately capture have been extensively explored, a comprehensive definition of the conductivity properties of these systems. Most relevantfor many-body localisation is still lacking. the purpose of this review are the transport properties of sim- ple paradigmatic models such as the Heisenberg spin chain or Dynamics of quantum phase transitions Hubbard-type models. In these models, transport is expected to lead to equilibration, as it allows the system to locally forget Exploring the dynamical signatures of phase transitions is its precise initial configuration6,70. In spin chains, transport an interesting problem in its own right. It connects to the im- typically refers to a spreading of magnetised domains71. In portant question of how slow an experimental ramp needs to 4 be to avoid the creation of defects when preparing a quan- constituents83,98. Due to the involved Coulomb interaction, tum phase and whether this is at all possible in the thermody- these systems are, among many other applications, well suited namic limit81. A key theoretical model for these transitions, to explore the validity of Lieb-Robinson bounds and their based on the adiabatic theorem, has been provided by Kibble breakdown for long-range interactions57,69. Photonic archi- and Zurek30 for the important case of moving from a gapped tectures as well as arrays of superconducting qubits offer fur- phase into criticality. Based solely on the critical exponents of ther promising platforms. While the precise computational the model, this strikingly simple formalism predicts the num- complexity of analogue simulators has not been identified yet, ber of introduced defects as a function of the velocity of the and while it is unclear to what extent genuinely reliable ana- ramp. It has originally been developed for thermal transitions, logue simulation is possible in the absence of fully-fledged where it has also been experimentally tested82–84. For quan- quantum error correction, there is evidence that such ana- tum phase transitions such a scaling can be derived in the limit logue simulators have the potential of outperforming classi- of infinitely slow ramps, based on universality arguments to- cal computers. Quantum many-body dynamics can already gether with adiabatic perturbation theory81 and seems to be be probed in experimental settings under precisely controlled capable of accurately describing the dynamics of quantum conditions for longer times than can be kept track of using phase transitions when crossing a single critical point in some state-of-the-art tensor network methods on modern classical models30. For strongly correlated systems in realistic experi- supercomputers14. mental settings, the dynamics of quantum phase transitions is more elusive and there are indications that the complexity of Periodically driven systems the dynamics cannot fully be captured by employing simple scaling arguments based only on the critical exponents of the 31,85 Periodically driving a well controlled quantum system model . The transition out of criticality has also been in- opens up new vistas for quantum simulations. Consider a vestigated, both numerically86 and experimentally32. Despite system with a Hamiltonian that satisfies H(t) = H(t + τ) these first promising results, the dynamics of quantum phase for all times t for some period τ > 0. The eigenstates of transitions and how to capture them in terms of simple scaling such systems are usually constantly changing in time, but the laws is still a largely open problem. long time dynamics can be inferred from their Floquet opera- tor99,100 U(τ), which is the time evolution operator over one Quantum simulations period of driving. The Floquet operator gives rise to an effec- tive Hamiltonian Heff via When investigating the non-equilibrium behaviour of large- scale many-body systems, analytical and numerical tools U(τ)= e−iτHeff , (6) quickly reach their limits, especially for lattice models in more than one dimension. A particularly exciting perspective arises and the time evolution of the driven system can be seen as from the idea that controlled quantum many-body systems a stroboscopic simulation101,102 of the time evolution under 14,31,87,88 constitute instances of analogue quantum simulators Heff . The effective Hamiltonian can be tuned by choosing overcoming these limitations. These devices mimic natural in- an appropriate driving schedule with great flexibility concern- teracting quantum many-body systems by reconstructing their ing the model parameters. This allows the study of topolog- Hamiltonian, but now under precisely controlled conditions. ical effects102 in driven systems and can be used to engineer This is in contrast to digital quantum simulators, anticipated gauge fields101. This has lead to an experimental realisation devices that approximatequantum dynamics by suitable quan- of the topological Haldane model in a periodically modulated tum circuits. Such analogue systems enable one to investigate optical honeycomb lattice103 as well as to experimental se- some of the questions central to the research summarised in tups featuring Hofstadter bands104 and spin-orbit coupling105 this review. in optical lattice architectures106. The synchronisation of the Ultra-cold atoms in optical lattices provide a very motion of periodically driven systems with the drive allows promising platform for quantum simulators in this the description of the long time behavior by a periodic ensem- sense89. With present technology they can be used to ble. For free systems it turns out to be given by a maximum 107 investigate dynamics under Bose- and Fermi-Hubbard entropy ensemble given the constants of motion — remi- Hamiltonians2,14,31,32,58,72,73,85,90–92 as well as for emulating niscent of Jaynes’ principle for systems in equilibrium and the spin chains93. Due to the flexibility of optical lattice sys- generalised Gibbs ensemble emerging during equilibration af- tems, e.g., by making use of Feshbach resonances, periodic ter a quench. driving, and superlattices, even sophisticated quantum simulations are conceivable, e.g., the simulation of lattice Perspectives for open and closed systems gauge theories94. Continuous optical setups allow for the investigation of out-of-equilibrium behaviour of quantum The research field of quantum many-body systems out of fields95 and have successfully been employed to investigate equilibrium is rapidly developing and is taking new and un- (pre-)thermalisation25 and out of equilibrium dynamics not expected directions at a remarkably high rate. This is par- leading to equilibration on relevant time scales96. tially driven by the impetus the field is subjected to through Another promising architecture is provided by ion novel experimental developments. In optical lattice systems, chains97, allowing for the precise control of the individual single-site addressing32,58,108 opens up new possibilities for 5 probing the out of equilibrium dynamics; settings with pe- very recently45,53–55,61,115. The conditions for thermalisation riodic driving now allow for the study of topologically non- are less clear and it is questionable whether (non-)integrability trivial situations102 and to develop a deeper understanding of is able to capture this phenomenonin all of its facets17,49. The particle physics and condensed matter systems101. Continuous problem of many-body localisation has recently moved in the systems can be controlled with ever increasing precision25,95, focus of attention79,80, but it seems fair to say that in the light suggesting that such systems can serve as a platform for prob- of the various pictures of the phenomenon in the literature, a ing even effects of gravity, such as the Unruh effect109,110. unifying framework is still missing. Hybrid architectures, such as cold atoms in optical lattices in- A research area that deserves special attention is based on side an optical cavity, constitute a particularly promising av- the idea that noise and dissipation are not necessarily detri- enue for experimental research111. Crucial for further progress mental to attempts to prepare and preserve interesting states will be a deeper understanding of the computational capa- of matter. In fact such preparations can be facilitated by suit- bility of analogue quantum simulators and how their correct ably engineered local Markovian noise116 that can be rigor- functioning can be certified in regimes that are classically not ously captured in terms of a Liouvillian. In this mindset, an addressable14,31,88,90. open system quantum simulator with trapped ions has been At the same time, new theoretical developments keep the realised117, as well as one based of cold atoms controlled subject exciting, with quantum information theory providing with localized dissipation118. In principle, even symmetry- a fresh point of view and new methods. Entanglement theory protected topologically ordered states can be prepared by can deliver important new insights specifically for the study purely dissipative local Liouvillian dynamics119. of information propagation56,59–61. Quantum thermodynam- It goes without saying that this perspectives article merely ics112 — exploring the ultimate limits of thermodynamic pro- touches the surface of this exciting and rapidly developing cesses in the quantum regime — is now seeking contact with field. Quantum many-body systems out of equilibrium al- this field113, with interesting perspectives opening up. The low us to probe long-standing questions in the foundations of AdS-cft-correspondence also contributes a novel tool to study statistical mechanics, relate to open questions in condensed- strongly correlated quantum systems out of equilibrium114. matter theory, and provide a promising platform for analogue In spite of the great progress, many questions are still wide quantum simulators outperforming classical computers. open, even very basic ones. While it is understood under We would like to thank many colleagues for numerous dis- which conditions many-body systems equilibrate, it is far cussions over the years, and E. Bergholtz, E. T. Campbell, from clear on what time scales they do so. The main reason A. del Campo, J.-S. Caux, F. Essler, T. Farrelly, M. Rigol, for this is that the role of locality for equilibration and ther- and R. Moessner for comments on the manuscript, and the malisation is, at least from a rigorous perspective, far from EU (RAQUEL, SIQS), the ERC, the BMBF, and the Studien- being understood. Steps in this direction have been taken only stiftung des Deutschen Volkes for support.

1 v. Neumann, J. Beweis des Ergodensatzes und des H-Theorems 10 Barthel, T., & Schollw¨ock, U. Dephasing and the steady state in der neuen Mechanik. Z. Phys. 57, 30 (1929). in quantum many-particle systems. Phys. Rev. Lett. 100, 100601 2 Greiner, M., Mandel, O., H¨ansch, T. W. & Bloch, I. Collapse and (2008). revival of the matter wave field of a Bose-Einstein condensate. 11 Manmana, S. R., Wessel, S., Noack, R. M. & Muramatsu, A. Nature 419, 51 (2002). Time evolution of correlations in strongly interacting fermions 3 Calabrese, P. & Cardy, J. Time dependence of correlation func- after a quantum quench. Phys. Rev. B 79, 155104 (2009). tions following a quantum quench. Phys. Rev. Lett. 96, 136801 12 Polkovnikov, A., Sengupta, K., Silva, A. & Vengalattore, M. (2006). Non-equilibrium dynamics of closed interacting quantum sys- 4 Kollath, C., L¨auchli, A. & Altman, E. Quench dynamics and non tems. Rev. Mod. Phys. 83, 863 (2011). equilibrium phase diagram of the Bose-Hubbard model. Phys. 13 Calabrese, P., Essler, F. H. L. & Fagotti, M. Quantum quench Rev. Lett. 98, 180601 (2007). in the transverse-field Ising chain. Phys. Rev. Lett. 106, 227203 5 Rigol, M., Dunjko, V., Yurovsky, V. & Olshanii, M. Relaxation (2011). in a completely integrable many-body quantum system: An ab 14 Trotzky, S. et al. Probing the relaxation towards equilibrium in initio study of the dynamics of the highly excited states of 1D an isolated strongly correlated one-dimensional Bose gas. Nature lattice hard-core bosons. Phys. Rev. Lett. 98, 050405 (2007). Phys. 8, 325 (2012). 6 Cramer, M., Dawson, C. M., Eisert, J. & Osborne, T. J. Quench- 15 Caux, J.-S. & Essler, F. H. L. Time evolution of local observables ing, relaxation, and a central limit theorem for quantum lattice after quenching to an integrable model. Phys. Rev. Lett. 110, systems. Phys. Rev. Lett. 100, 030602 (2008). 257203 (2013). 7 Flesch, A., Cramer, M., McCulloch, I. P., Schollw¨ock, U. & Eis- 16 Barmettler, P., Punk, M., Gritsev, V., Demler, E. & Altman, E. 1 ert, J. Probing local relaxation of cold atoms in optical superlat- Relaxation of antiferromagnetic order in spin- 2 chains following tices. Phys. Rev. A 78, 033608 (2008). a quantum quench. Phys. Rev. Lett. 102, 130603 (2009). 8 Moeckel, M. & and Kehrein, S. Interaction quench in the Hub- 17 Gogolin, C., Mueller, M. P. & Eisert, J. Absence of thermal- bard model. Phys. Rev. Lett. 100, 175702 (2008). ization in non-integrable systems. Phys. Rev. Lett. 106, 040401 9 Rigol, M., Dunjko, V. & Olshanii, M. Thermalization and its (2011). mechanism for generic isolated quantum systems. Nature 452, 18 Cassidy, A. C., Clark, C. W. & Rigol, M. Generalized thermaliza- 854 (2008). tion in an integrable lattice system. Phys. Rev. Lett. 106, 140405 6

(2011). 112, 130403 (2014). 19 Cramer, M. & Eisert, J. A quantum central limit theorem for non- 43 Beugeling, W., Moessner, R. & Haque, M. Finite-size scaling of equilibrium systems: Exact local relaxation of correlated states. eigenstate thermalization. Phys. Rev. E 89, 042112 (2014). New J. Phys. 12, 055020 (2010). 44 Riera, A., Gogolin, C. & Eisert, J. Thermalization in nature and 20 Reimann, P. Foundation of statistical mechanics under exper- on a quantum computer. Phys. Rev. Lett. 108, 080402 (2012). imentally realistic conditions. Phys. Rev. Lett. 101, 190403 45 Mueller, M. P., Adlam, E., Masanes, L. & Wiebe, N. Thermal- (2008). ization and canonical typicality in translation-invariant quantum 21 Linden, N., Popescu, S., Short, A. J. & Winter, A. Quantum lattice systems. ArXiv:1312.7420. mechanical evolution towards thermal equilibrium. Phys. Rev. E 46 del Rio, L., Hutter, A., Renner, R. & Wehner, S. Relative ther- 79, 061103 (2009). malization (2014). ArXiv:1401.7997. 22 Reimann, P. & Kastner, M. Equilibration of isolated macroscopic 47 Goldstein, S. Canonical typicality. Phys. Rev. Lett. 96, 050403 quantum systems. New J. Phys. 14, 043020 (2012). (2006). 23 Short, A. J. & Farrelly, T. C. Quantum equilibration in finite time. 48 Popescu, S., Short, A. J. & Winter, A. The foundations of statisti- New J. Phys. 14, 013063 (2012). cal mechanics from entanglement: Individual states vs. averages. 24 Langen, T., Geiger, R., Kuhnert, M., Rauer, B. & Schmiedmayer, Nature Phys. 2, 754 (2006). J. Local emergence of thermal correlations in an isolated quan- 49 Caux, J.-S. & Mossel, J. Remarks on the notion of quantum tum many-body system. Nature Phys. 9, 640 (2013). integrability. J. Stat. Mech. P02023 (2011). 25 Geiger, R., Langen, T., Mazets, I. & Schmiedmayer, J. Local 50 Altland, A. & Haake, F. Quantum chaos and effective thermal- relaxation and light-cone-like propagation of correlations in a ization. Phys. Rev. Lett. 108, 073601 (2012). trapped one-dimensional Bose gas. New. J. Phys. 16, 053034 51 Marcuzzi, M., Marino, J., Gambassi, A. & Silva, A. Prethermal- (2014). ization in a non-integrable quantum spin chain after a quench. 26 Torres-Herrera, E. J., Kollmar, D. & Santos, L. F. Relax- Phys. Rev. Lett. 111, 197203 (2013). ation and thermalization of isolated many-body quantum sys- 52 Essler, F. H. L., Kehrein, S., Manmana, S. R., Robinson, N. J., tems. ArXiv:1403.6481. Quench dynamics in a model with tuneable integrability break- 27 Venuti, L. C. & Zanardi, P. Universal time-fluctuations in near- ing. Phys. Rev. B 89, 165104 (2014). critical out-of-equilibrium quantum dynamics. Phys. Rev. E 89, 53 Hastings, M. B. & Koma, T. Spectral gap and exponential decay 022101 (2014). of correlations. Commun. Math. Phys. 265, 781 (2006). 28 Torres-Herrera, E. J. & Santos, L. F. Local quenches with global 54 Sidoravicius, V. (ed.). Locality estimates for quantum spin sys- effects in interacting quantum systems. Phys. Rev. E 89, 062110 tems (Springer Verlag, 2009). (2014). 55 Bravyi, S., Hastings, M. B. & Verstraete, F. Lieb-Robinson 29 Stephan, J.-M. & Dubail, J. Local quantum quenches in critical bounds and the generation of correlations and topological quan- one-dimensional systems: Entanglement, the Loschmidt echo, tum order. Phys. Rev. Lett. 97, 050401 (2006). and light-cone effects. J. Stat. Mech. P08019 (2011). 56 Calabrese, P. & Cardy, J. Evolution of entanglement entropy in 30 del Campo, A. & Zurek, W. H. Universality of phase transition one-dimensional systems. J. Stat. Mech. P04010 (2005). dynamics: Topological defects from symmetry breaking. Int. J. 57 Jurcevic, P. et al. Quasiparticle engineering and entanglement Mod. Phys. A 29, 1430018 (2014). propagation in a quantum many-body system. Nature 511, 202 31 Braun, S. et al. Emergence of coherence and the dynamics of (2014). quantum phase transitions. ArXiv:1403.7199. 58 Cheneau, M. et al. Light-cone-like spreading of correlations in a 32 Bakr, W. S. et al. Probing the superfluid-to-Mott insulator transi- quantum many-body system. Nature 481, 484 (2012). tion at the single-atom level. Science 329, 547 (2010). 59 Hastings, M. B. An area law for one-dimensional quantum sys- 33 D´ora, B., Haque, M. & Zar´and, G. Crossover from adiabatic to tems. J. Stat. Mech. P08024 (2007). sudden interaction quench in a Luttinger liquid. Phys. Rev. Lett. 60 Eisert, J., Cramer, M. & Plenio, M. B. Area laws for the entan- 106, 156406 (2011). glement entropy. Rev. Mod. Phys. 82, 277 (2010). 34 Mossel, J., Palacios, G. & Caux, J.-S. Geometric quenches in 61 Eisert, J. & Osborne, T. J. General entanglement scaling laws quantum integrable systems. J. Stat. Mech. L09001 (2010). from time evolution. Phys. Rev. Lett 97, 150404 (2006). 35 Alba, V. & Heidrich-Meisner, F. Entanglement spreading after a 62 Schuch, N., Wolf, M. M., Vollbrecht, K. G. H. & Cirac, J. I. On geometric quench in quantum spin chains. ArXiv:1402.2299. entropy growth and the hardness of simulating time evolution. 36 Bovensiepen, U., Petek, H. & Wolf, M. (eds.) Dynamics at solid New J. Phys. 10, 033032 (2008). state surfaces and interfaces (Wiley-VCH, Weinheim, Germany, 63 Orus, R. A practical introduction to tensor networks: Matrix 2010). product states and projected entangled pair states. Ann. Phys. 37 Deutsch, J. M. Quantum statistical mechanics in a closed system. 349, 117 (2014). Phys. Rev. A 43, 2046 (1991). 64 Banuls, M. C., Hastings, M. B., Verstraete, F. & Cirac, J. I. Ma- 38 Srednicki, M. Chaos and quantum thermalization. Phys. Rev. E trix product states for dynamical simulation of infinite chains. 50, 888 (1994). Phys. Rev. Lett. 102, 240603 (2009). 39 Tasaki, H. From quantum dynamics to the canonical distribution: 65 Schollw¨ock, U. The density-matrix renormalization group in the General picture and a rigorous example. Phys. Rev. Lett. 80, 1373 age of matrix product states. Ann. Phys. 326, 96 (2011). (1998). 66 Troyer, M., Alet, F., Trebst, S. & Wessel, S. Non-local updates 40 Rigol, M. & Srednicki, M. Alternatives to eigenstate thermaliza- for quantum Monte Carlo simulations. AIP Conf. Proc. 690, 156 tion. Phys. Rev. Lett. 108, 110601 (2012). (2003). 41 Rigol, M. Breakdown of thermalization in finite one-dimensional 67 Aoki, H. et al. Non-equilibrium dynamical mean-field theory and systems. Phys. Rev. Lett. 103, 100403 (2009). its applications. Rev. Mod. Phys. 86, 779 (2014). 42 Steinigeweg, R., Khodja, A., Niemeyer, H., Gogolin, C. & Gem- 68 Nagaj, D. Fast universal quantum computation with railroad- mer, J. Pushing the limits of the eigenstate thermalization hy- switch local Hamiltonians. J. Math. Phys. 51, 62201 (2010). pothesis towards mesoscopic quantum systems. Phys. Rev. Lett. 69 Richerme, P. et al. Non-local propagation of correlations in long- 7

range interacting quantum systems (2014). ArXiv:1401.5088. ising chain. Phys. Rev. Lett. 111, 053003 (2013). 70 Gogolin, C. & Eisert, J. Pure state statistical mechanics. In prepa- 94 Zohar, E., Cirac, J. I. & Reznik, B. A cold-atom quantum sim- ration (2014). ulator for su(2) Yang-Mills lattice gauge theory. Phys. Rev. Lett. 71 Langer, S., Heidrich-Meisner, F., Gemmer, J., McCulloch, I. P. & 110, 125304 (2013). Schollw¨ock, U. Real-time study of diffusive and ballistic trans- 95 Hofferberth, S., Lesanovsky, I., Fischer, B., Schumm, T. & 1 port in spin- 2 chains using the adaptive time-dependent density Schmiedmayer, J. Non-equilibrium coherence dynamics in one- matrix renormalization group method. Phys. Rev. B 79, 214409 dimensional Bose gases. Nature 449, 324 (2007). (2009). 96 Kinoshita, T., Wenger, T. & Weiss, D. S. A quantum Newton’s 72 Schneider, U. et al. Fermionic transport and out-of-equilibrium cradle. Nature 440, 900 (2006). dynamics in a homogeneous Hubbard model with ultra-cold 97 Blatt, R. & Roos, C. F. Quantum simulations with trapped ions. atoms. Nat. Phys. 8, 213 (2012). Nature Phys. 8, 277 (2012). 73 Ronzheimer, J. P. et al. Expansion dynamics of interacting 98 Islam, R. et al. Onset of a quantum phase transition with a bosons in homogeneous lattices in one and two dimensions. trapped ion quantum simulator. Nat. Commun. 2, 377 (2011). Phys. Rev. Lett. 110, 205301 (2013). 99 Maricq, M. M. Application of average Hamiltonian theory to the 74 Schuch, N., Wolf, M. M., Verstraete, F. & Cirac, J. I. Entropy NMR of solids. Phys. Rev. B 25, 6622 (1982). scaling and simulability by matrix product states. Phys. Rev. Lett. 100 Arimondo, E., Ciampini, D., Eckardt, A., Holthaus, M. & 100, 030504 (2008). Morsch, O. Kilohertz-driven Bose-Einstein condensates in op- 75 Lagendijk, A., van Tiggelen, B. & Wiersma, D. S. Fifty years of tical lattices. Adv. At. Mol. Opt. Phys. 61, 515 (2012). Anderson localization. Physics Today 62, 24 (2009). 101 Goldman, N. & Dalibard, J. Periodically-driven quantum 76 Bardarson, J. H., Pollmann, F. & Moore, J. E. Unbounded growth systems: Effective Hamiltonians and engineered gauge fields. of entanglement in models of many-body localization. Phys. Rev. ArXiv:1404.4373. Lett. 109, 017202 (2012). 102 Kitagawa, T., Berg, E., Rudner, M. & Demler, E. Topological 77 Hamza, E., Sims, R. & Stolz, G. Dynamical localization in dis- characterization of periodically driven quantum systems. Phys. ordered quantum spin systems. Commun. Math. Phys. 315, 215 Rev. B 82, 235114 (2010). (2012). 103 Jotzu, G. et al. Experimental realization of the topological Hal- 78 Aleiner, I. L., Altshuler, B. L. & Shlyapnikov, G. V. A finite- dane model. ArXiv:1406.7874. temperature phase transition for disordered weakly interacting 104 Aidelsburger, M. et al. Revealing the topology of Hofstadter bosons in one dimension. Ann. Phys. 321, 1126 (2006). bands with ultra-cold atoms. ArXiv:1407.4205. 79 Nandkishore, R. & Huse, D. A. Many body localization and ther- 105 Struck, J., Simonet, J. & Sengstock, K. Spin orbit coupling in malization in quantum statistical mechanics. ArXiv:1404.0686. periodically driven optical lattices. ArXiv:1407.1953. 80 Bauer, B. & Nayak, C. Area laws in a many-body localized state 106 Goldman, N. et al. Direct imaging of topological edge states in and its implications for topological order. J. Stat. Mech. P09005 cold-atom systems. PNAS 110, 6736 (2013). (2013). 107 Lazarides, A., Das, A. & Moessner, R. Periodic thermodynamics 81 Polkovnikov, A. & Gritsev, V. Breakdown of the adiabatic limit of isolated systems. Phys. Rev. Lett. 112, 150401 (2014). in low-dimensional gapless systems. Nature Phys. 4, 477 (2008). 108 Endres, M. et al. Single-site- and single-atom-resolved measure- 82 Ruutu, V. M. H. et al. Vortex formation in neutron-irradiated ment of correlation functions. ArXiv:1303.5652. superfluid 3He as an analogue of cosmological defect formation. 109 Martin-Martinez, E., Fuentes, I. & Mann, R. B. Using Berry’s Nature 382, 334 (1996). phase to detect the Unruh effect at lower accelerations. Phys. 83 Ulm, S. et al. Observation of the Kibble-Zurek scaling law for Rev. Lett. 107, 131301 (2011). defect formation in ion crystals. Nat. Comm. 4, 2290 (2013). 110 Agarwal, K. et al. Chiral prethermalization in supersonically split 84 Pyka, K. et al. Topological defect formation and spontaneous condensates. ArXiv: 1402.6716. symmetry breaking in ion coulomb crystals. Nat. Comm. 4 111 Ritsch, H., Domokos, P., Brennecke, F. & Esslinger, T. Cold (2013). atoms in cavity-generated dynamical optical potentials. Rev. 85 Chen, D., White, M., Borries, C. & DeMarco, B. Quantum Mod. Phys. 85, 553 (2013). quench of an atomic Mott insulator. Phys. Rev. Lett. 106, 235304 112 Linden, N., Popescu, S. & Skrzypczyk, P. How small can thermal (2011). machines be? The smallest possible refrigerator, Phys. Rev. Lett. 86 Bernier, J.-S., Poletti, D., Barmettler, P., Roux, G. & Kollath, C. 105, 130401 (2010). Slow quench dynamics of Mott-insulating regions in a trapped 113 Gallego, R., Riera, A. & Eisert, J. Correlated thermal machines Bose gas. Phys. Rev. A 85, 033641 (2012). in the micro-world (2013). Arxiv.1310.8349. 87 Cirac, J. I. & Zoller, P. Goals and opportunities in quantum sim- 114 Hubeny, V. E. & Rangamani, M. A holographic view on physics ulation. Nature Phys. 8, 264 (2012). out of equilibrium. Adv. High En. Phys. 2010, 297916 (2010). 88 Hauke, P., Cucchietti, F. M., Tagliacozzo, L., Deutsch, I. & 115 Kliesch, M., Gogolin, C., Kastoryano, M. J., Riera, A. & Eisert, Lewenstein, M. Can one trust quantum simulators? Rep. Prog. J. Locality of temperature. Phys. Rev. X 4, 031019 (2014). Phys. 75, 082401 (2012). 116 Diehl, S. et al. Quantum states and phases in driven open quan- 89 Bloch, I., Dalibard, J. & Nascimbene, S. Quantum simulations tum systems with cold atoms. Nature Phys. 4, 878 (2008). with ultracold quantum gases. Nature Phys. 8, 267 (2012). 117 Barreiro, J. T. et al. An open-system quantum simulator with 90 Trotzky, S. et al. Suppression of the critical temperature for su- trapped ions. Nature 470, 486 (2011). perfluidity near the Mott transition: validating a quantum simu- 118 Barontini, G. et al. Controlling the dynamics of an open many- lator. Nature Phys. 6, 998 (2010). body quantum system with localized dissipation. Phys. Rev. Lett. 91 Esslinger, T. Fermi-Hubbard physics with atoms in an optical 110, 035302 (2013). lattice. Ann. Rev. Cond. Mat. Phys. 1, 129 (2010). 119 Diehl, S., Rico, E., Baranov, M. A. & Zoller, P. Topology by 92 Pertot, D. et al. Relaxation dynamics of a Fermi gas in an optical dissipation in atomic quantum wires. Nature Phys. 7, 971 (2011). superlattice. ArXiv:1407.6037. 93 Meinert, F. et al. Quantum quench in an atomic one-dimensional