Long-time Behavior of Isolated Periodically Driven Interacting Lattice Systems

D’Alessio, Luca; Rigol, Marcos

doi:10.1103/physrevx.4.041048

Cited by 585 publications

(548 citation statements)

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“…The robust subharmonic synchronization of the many-body Floquet system is the essence of the discrete time crystal phase [7][8][9][10] . In a DTC, the underlying Floquet drive should generally be accompanied by strong disorder, leading to manybody localization 13 and thereby preventing the quantum system from absorbing the drive energy and heating to infinite temperatures [14][15][16][17] .…”

mentioning

confidence: 99%

Observation of a discrete time crystal

Zhang

Hess

Kyprianidis

et al. 2017

Nature

1,111

972

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Spontaneous symmetry breaking is a fundamental concept in many areas of physics, ranging from cosmology and particle physics to condensed matter 1 . A prime example is the breaking of spatial translation symmetry, which underlies the formation of crystals and the phase transition from liquid to solid. Analogous to crystals in space, the breaking of translation symmetry in time and the emergence of a "time crystal" was recently proposed 2,3 , but later shown to be forbidden in thermal equilibrium [4][5][6] . However, nonequilibrium Floquet systems subject to a periodic drive can exhibit persistent time-correlations at an emergent sub-harmonic frequency [7][8][9][10] . This new phase of matter has been dubbed a "discrete time crystal" (DTC) 10,11 . Here, we present the first experimental observation of a discrete time crystal, in an interacting spin chain of trapped atomic ions. We apply a periodic Hamiltonian to the system under many-body localization (MBL) conditions, and observe a sub-harmonic temporal response that is robust to external perturbations. Such a time crystal opens the door for studying systems with long-range spatial-temporal correlations and novel phases of matter that emerge under intrinsically non-equilibrium conditions 7 .For any symmetry in a Hamiltonian system, its spontaneous breaking in the ground state leads to a phase transition 12 . The broken symmetry itself can assume many different forms. For example, the breaking of spinrotational symmetry leads to a phase transition from paramagnetism to ferromagnetism when the temperature is brought below the Curie point. The breaking of spatial symmetry leads to the formation of crystals, where the continuous translation symmetry of space is replaced by a discrete one.We now pose an analogous question: can the translation symmetry of time be broken? The proposal of such a "time crystal" 2 for time-independent Hamiltonians has led to much discussion, with the conclusion that such structures cannot exist in the ground state or any thermal equilibrium state of a quantum mechanical system 4-6 . A simple intuitive explanation is that quantum equilibrium states have time-independent observables by construction; thus, time translation symmetry can only be spontaneously broken in non-equilibrium systems 7-10 . In particular, the dynamics of periodically-driven Floquet systems possesses a discrete time translation symmetry governed by the drive period. This symmetry can be further broken into "super-lattice" structures where physical observables exhibit a period larger than that of the drive. Such a response is analogous to commensurate charge density waves that break the discrete translation symmetry of their underlying lattice 1 . The robust subharmonic synchronization of the many-body Floquet system is the essence of the discrete time crystal phase 7-10 . In a DTC, the underlying Floquet drive should generally be accompanied by strong disorder, leading to manybody localization 13 and thereby preventing the quantum system from absorbing the drive energy...

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mentioning

confidence: 99%

Observation of a discrete time crystal

Zhang

Hess

Kyprianidis

et al. 2017

Nature

1,111

972

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“…The persistence of the critical behaviour for moderate interactions suggests the absence of many-body delocalization of the FGS. Although other parts of the Floquet spectrum may undergo many-body delocalization [56,57,[60][61][62]105], our method does not give the possibility to directly observe it. Our results were obtained by means of limited resources [100]: the resulting uncertainty does not allow us to clarify what happens in our system for large interactions.…”

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confidence: 99%

“…The persistence of the non-equilibrium transition is probably connected with its topological nature, which prevents these gaps from opening. We conjecture that in the presence of larger interactions a strong mixing of the Floquet states is expected, leading to Floquet states obeying Eigenstate Thermalization [20,49,50,62,65,67].…”

mentioning

confidence: 99%

“…Our results were obtained by means of limited resources [100]: the resulting uncertainty does not allow us to clarify what happens in our system for large interactions. Further work is therefore needed to see if the FGS itself can undergo manybody delocalization in our model and in other interesting 1d systems [56,57,[60][61][62]105]. * * * We acknowledge useful discussions with E. Berg, R. Berkovitz, E. Demler and R. Fazio.…”

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confidence: 99%

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Kibble-Zurek scaling in periodically driven quantum systems

Russomanno¹,

Torre²

2016

EPL

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-We study the slow crossing of non-equilibrium quantum phase transitions in periodically-driven systems. We explicitly consider a spin chain with a uniform time-dependent magnetic field and focus on the Floquet state that is adiabatically connected to the ground state of the static model. We find that this Floquet ground state undergoes a series of quantum phase transitions characterized by a non-trivial topology. To dynamically probe these transitions, we propose to start with a large driving frequency and slowly decrease it as a function of time. Combining analytical and numerical methods, we uncover a Kibble-Zurek scaling that persists in the presence of moderate interactions. This scaling can be used to experimentally demonstrate non-equilibrium transitions that cannot be otherwise observed.Recent experiments on artificial quantum many-body systems demonstrated quantum-coherent evolution under various external perturbations [1,[1][2][3][4] and triggered the study of fundamental aspects of quantum non-equilibrium physics. One important direction is the quest for universal properties in the dynamics [4][5][6][7][8][9][10][11]. A universal non-equilibrium behaviour can originate from the universality of an underlying equilibrium problem [5,6]: A well known example is offered by the Kibble-Zurek (K-Z) scaling observed when crossing a quantum phase transition [4,7,[12][13][14]. In addition, recent studies found evidence of genuine non-equilibrium universality following a quantum quench [8][9][10][11], or in presence of a timedependent noise [15,16]. Another important direction is the exploration of the conditions for an asymptotic steady state [4,. In the context of quantum quenches, it was found that systems with a continuous spectrum generically relax to a diagonal ensemble [37,38]. Current investigations focus on understanding when this asymptotic condition is thermal (ergodic dynamics) and when it is not (regular dynamics) [20,21,[23][24][25]39]. The transition from regular to ergodic dynamics is connected to the emergence of quantum chaos [41][42][43] and to the many-body localization/delocalization transition [24,25,39,[44][45][46][47][48]. These studies are deeply related to the main problem of statistical mechanics: how microscopic quantum coherent dynamics can induce macroscopic thermalization [20,39,49,50].In this work we address the universal scaling and the absence of thermalization in periodically-driven many-body systems . We focus on Floquet states, the eigenstates of the periodically driven dynamics. We specifically address a Floquet state which shows many properties analogous to the ground state in the static system: we term "Floquet ground state" (FGS). We emphasize that -opposite to the GS of a static Hamiltonian -the FGS depends on time in a periodic fashion (up to a phase factor) and, in general, cannot be found as the minimum-energy eigenstate of any Hamiltonian. Nevertheless it shows many properties similar to the GS of a static Hamiltonian. First of all, it shows an area-law entan...

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“…We mention that the parameter in the Hamiltonian of a closed system can be driven periodically or randomly in time, and the non-equilibrium dynamics of these driven systems draws attention both experimental [75] and theoretical [76] [77].…”

Section: Introductionmentioning

confidence: 99%

Non-equilibrium dynamics of one-dimensional isolated quantum systems

Roósz

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ii To Maria iii Szerzői nyilatkozatKijelentem, hogy a doktori disszertációmban foglaltak a 2.és a 3. fejezetek kivételével, melyek az irodalomban fellelhető eredményeket tekintikát, saját munkám eredményei,és csak a hivatkozott forrásokat használtam fel. Tudomásul veszem azt, hogy disszertációmat a Szegedi Tudományegyetem könyvtárában, a kölcsönözhető könyvek között helyezik el.

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Long-time Behavior of Isolated Periodically Driven Interacting Lattice Systems

Cited by 585 publications

References 72 publications

Observation of a discrete time crystal

Observation of a discrete time crystal

Kibble-Zurek scaling in periodically driven quantum systems

Non-equilibrium dynamics of one-dimensional isolated quantum systems

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