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arXiv:1704.01591 · Physical review. B./Physical review. B

Floquet time crystal in the Lipkin-Meshkov-Glick model

Angelo Russomanno, Fernando Iemini, Marcello Dalmonte, Rosario Fazio

Published Jun 22, 2017• 222 citations
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Abstract

In this work we discuss the existence of time-translation symmetry breaking in a kicked infinite-range-interacting clean spin system described by the Lipkin-Meshkov-Glick model. This Floquet time crystal is robust under perturbations of the kicking protocol, its existence being intimately linked to the underlying ${\mathbb{Z}}_{2}$ symmetry breaking of the time-independent model. We show that the model being infinite range and having an extensive amount of symmetry-breaking eigenstates is essential for having the time-crystal behavior. In particular, we discuss the properties of the Floquet spectrum, and show the existence of doublets of Floquet states which are, respectively, even and odd superposition of symmetry-broken states and have quasienergies differing of half the driving frequencies, a key essence of Floquet time crystals. Remarkably, the stability of the time-crystal phase can be directly analyzed in the limit of infinite size, discussing the properties of the corresponding classical phase space. Through a detailed analysis of the robustness of the time crystal to various perturbations we are able to map the corresponding phase diagram. We finally discuss the possibility of an experimental implementation by means of trapped ions.

DOI
10.1103/physrevb.95.214307

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In this work we discuss the existence of time-translation symmetry breaking in a kicked infinite-range-interacting clean spin system described by the Lipkin-Meshkov-Glick model. This Floquet time crystal is robust under perturbations of the kicking protocol, its existence being intimately linked to the underlying Z2{\mathbb{Z}}_{2}Z2​ symmetry breaking of the time-independent model. We show that the model being infinite range and having an extensive amount of symmetry-breaking eigenstates is essential for having the time-crystal behavior. In particular, we discuss the properties of the Floquet spectrum, and show the existence of doublets of Floquet states which are, respectively, even and odd superposition of symmetry-broken states and have quasienergies differing of half the driving frequencies, a key essence of Floquet time crystals. Remarkably, the stability of the time-crystal phase can be directly analyzed in the limit of infinite size, discussing the properties of the corresponding classical phase space. Through a detailed analysis of the robustness of the time crystal to various perturbations we are able to map the corresponding phase diagram. We finally discuss the possibility of an experimental implementation by means of trapped ions.
A discrete time is a remarkable non-equilibrium phase of matter characterized by persistent sub-harmonic response to a periodic drive. Motivated by the question of whether such time-crystalline order can persist when the drive becomes aperiodic, we investigate the dynamics of a Lipkin-Meshkov-Glick model under quasiperiodic kicking. Intriguingly, this infinite-range-interacting spin chain can exhibit long-lived periodic oscillations when the kicking amplitudes are drawn from the Thue-Morse sequence (TMS). We dub this phase a ``self-ordered time crystal (SOTC), and demonstrate that our model hosts at least two qualitatively distinct prethermal SOTC phases. These SOTCs are robust to various perturbations, and they originate from the interplay of long-range interactions and the recursive structure of the TMS. Our results suggest that quasiperiodic driving protocols can provide a promising route for realizing novel non-equilibrium phases of matter in long-range interacting systems.
Recent works on the observation of discrete time-crystalline signatures throw up major puzzles on the necessity of localization for stabilizing such out-of-equilibrium phases. Motivated by these studies, we delve into a clean interacting Floquet system whose quasi-spectrum conforms to the ergodic Wigner-Dyson distribution, yet with an unexpectedly robust, long-lived time-crystalline dynamics in the absence of disorder or fine-tuning. We relate such behavior to a measure zero set of nonthermal Floquet eigenstates with long-range spatial correlations, which coexist with otherwise thermal states at near-infinite temperature and develop a high overlap with a family of translationally invariant, symmetry-broken initial conditions. This resembles the notion of ``dynamical scars'' that remain robustly localized throughout a thermalizing Floquet spectrum with fractured structure. We dub such a long-lived discrete time crystal formed in partially nonergodic systems, ``scarred discrete time crystal'' which is distinct by nature from those stabilized by either many-body localization or prethermalization mechanism.
Despite being forbidden in equilibrium, spontaneous breaking of time translation symmetry can occur in periodically driven, Floquet systems with discrete time-translation symmetry.The period of the resulting discrete time crystal is quantized to an integer multiple of the drive period, arising from a combination of collective synchronization and many body localization.Here, we consider a simple model for a onedimensional discrete time crystal which explicitly reveals the rigidity of the emergent oscillations as the drive is varied.We numerically map out its phase diagram and compute the properties of the dynamical phase transition where the time crystal melts into a trivial Floquet insulator.Moreover, we demonstrate that the model can be realized with current experimental technologies and propose a blueprint based upon a one dimensional chain of trapped ions.Using experimental parameters (featuring long-range interactions), we identify the phase boundaries of the ion-time-crystal and propose a measurable signature of the symmetry breaking phase transition.
We construct a class of period-nnn-tupling discrete time crystals based on Zn{\mathbb{Z}}_{n}Zn​ clock variables, for all the integers nnn. We consider two classes of systems where this phenomenology occurs: disordered models with short-range interactions and fully connected models. In the case of short-range models, we provide a complete classification of time-crystal phases for generic nnn. For the specific cases of n=3n=3n=3 and n=4n=4n=4, we study in detail the dynamics by means of exact diagonalization. In both cases, through an extensive analysis of the Floquet spectrum, we are able to fully map the phase diagram. In the case of infinite-range models, the mapping onto an effective bosonic Hamiltonian allows us to investigate the scaling to the thermodynamic limit. After a general discussion of the problem, we focus on n=3n=3n=3 and n=4n=4n=4, representative examples of the generic behavior. Remarkably, for n=4n=4n=4 we find clear evidence of a crystal-to-crystal transition between period nnn-tupling and period n/2n/2n/2-tupling.
Time crystals correspond to a phase of matter where time-translational symmetry (TTS) is broken. Up to date, they are well studied in open quantum systems, where an external drive allows us to break discrete TTS, ultimately leading to Floquet time crystals. At the same time, genuine time crystals for closed quantum systems are believed to be impossible. In this study we propose a form of a Hamiltonian for which the unitary dynamics exhibits the time crystalline behavior and breaks continuous TTS. This is based on a spin-1/2 many-body Hamiltonian which has long-range multispin interactions in the form of spin strings, thus bypassing previously known no-go theorems. We show that quantum time crystals are stable to local perturbations at zero temperature. Finally, we reveal the intrinsic connection between continuous and discrete TTS, thus linking the two realms.
Despite being forbidden in equilibrium, spontaneous breaking of time translation symmetry can occur in periodically driven, Floquet systems with discrete time-translation symmetry. The period of the resulting discrete time crystal is quantized to an integer multiple of the drive period, arising from a combination of collective synchronization and many body localization. Here, we consider a simple model for a one-dimensional discrete time crystal which explicitly reveals the rigidity of the emergent oscillations as the drive is varied. We numerically map out its phase diagram and compute the properties of the dynamical phase transition where the time crystal melts into a trivial Floquet insulator. Moreover, we demonstrate that the model can be realized with current experimental technologies and propose a blueprint based upon a one dimensional chain of trapped ions. Using experimental parameters (featuring long-range interactions), we identify the phase boundaries of the ion-time-crystal and propose a measurable signature of the symmetry breaking phase transition.