Observation of discrete time-crystalline order in a disordered dipolar many-body system

Choi, Soonwon; Choi, Joonhee; Landig, Renate; Kucsko, Georg; Zhou, Hao; Isoya, Junichi; Jelezko, Fedor; Onoda, Shinobu; Sumiya, Hitoshi; Khemani, Vedika; Keyserlingk, C. W. von; Yao, Norman Y.; Demler, Eugene; Lukin, Mikhail D.

doi:10.1038/nature21426

Cited by 932 publications

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“…This includes the observation of genuine nonequilibrium phenonema such as many-body localization [1][2][3], quantum time crystals [4,5], or particle-antiparticle production in the Schwinger model [6]. It remains, however, a major challenge to identify universal properties in these diverse dynamical phenomena on general grounds.…”

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confidence: 99%

Connecting dynamical quantum phase transitions and topological steady-state transitions by tuning the energy gap

Wang

Xianlong

2018

Phys. Rev. A

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Considerable theoretical and experimental efforts have been devoted to the quench dynamics, in particular, the dynamical quantum phase transition (DQPT) and the steady-state transition. These developments have motivated us to study the quench dynamics of the topological systems, from which we find the connection between these two transitions, that is, the DQPT, accompanied by a nonanalytic behavior as a function of time, always merges into a steady-state transition signaled by the nonanalyticity of observables in the steady limit. As the characteristic time of the DQPT diverges, it exhibits universal scaling behavior, which is related to the scaling behavior at the corresponding steady-state transition.Isolated quantum many-body systems can nowadays be realized in quantum-optical systems such as ultracold atoms or trapped ions which has opened up the perspective to experimentally study properties beyond the thermodynamic equilibrium paradigm. This includes the observation of genuine nonequilibrium phenonema such as many-body localization [1][2][3], quantum time crystals [4,5], or particle-antiparticle production in the Schwinger model [6]. It remains, however, a major challenge to identify universal properties in these diverse dynamical phenomena on general grounds. Considerable effects have been made to the formulation of various notions of nonequilibrium phase transitions [7][8][9][10][11][12][13][14][15][16][17] which are seen as promising attempts to extend elementary equilibrium concepts such as scaling and universality to the nonequilibrium regime. Among these notions there is the concept of a steady-state transition, which is signaled by a nonanalytic change of physical properties as a function of a parameter of the nonequilibrium protocol in the asymptotic long-time state of the system [8,11,12]. An example is the universal logarithmic divergence of the Hall conductance in the steady state of topological insulators after a quench [16,17]. Another important concept is that of dynamical quantum phase transitions (DQPTs) [15], recently observed experimentally [18,19], which occur on transient and intermediate time scales accompanied by a nonanalytic behavior as a function of time instead of a conventional order parameter. In the context of such developments, it is then a natural question to ask, whether and how these two notions of phase transitions are connected.The DQPT and the steady-state transition under the symmetry-breaking picture have been recently connected in the long-range-interacting Ising chain [20] by relating the singularities of Loschmidt echo to the zeros of local order parameters. In this work, we study the connection between the DQPT and the steady-state transition in a topological system where the local order parameters are absent. The connection is schematically displayed in Fig. 1. The characteristic time scale t * associated with DQPTs diverges when tuning the energy gap of the postquench Hamiltonian towards the steady-state transition at the gap-closing point. The steady-state trans...

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confidence: 99%

Connecting dynamical quantum phase transitions and topological steady-state transitions by tuning the energy gap

Wang

Xianlong

2018

Phys. Rev. A

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“…Oscillations with period 2t F due to simultaneously initialized protected boundary states were studied in photonic quantum walks [3]; period-two oscillations can also be expected from the coexistence of Floquet Majorana fermions with quasienergies 0 and π/t F in a cold-atom system [4]. The onset of periodtwo phases was predicted and analyzed [5][6][7][8][9][10] in Floquet many-body localized systems, and the first observations of oscillations at multiples of the driving period in disordered systems were reported [11,12].In systems coupled to a thermal bath, on the other hand, the effect of period doubling has been well-known. A textbook example is a classical oscillator modulated close to twice its eigenfrequency and displaying vibrations with period 2t F [13].…”

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confidence: 99%

Time-translation-symmetry breaking in a driven oscillator: From the quantum coherent to the incoherent regime

et al. 2017

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We study the breaking of the discrete time-translation symmetry in small periodically driven quantum systems. Such systems are intermediate between large closed systems and small dissipative systems, which both display the symmetry breaking, but have qualitatively different dynamics. As a nontrivial example we consider period tripling in a quantum nonlinear oscillator. We show that, for moderately strong driving, the period tripling is robust on an exponentially long time scale, which is further extended by an even weak decoherence.The breaking of translation symmetry in time, first proposed by Wilczek [1], has been attracting much attention recently. Such symmetry breaking can occur only away from thermal equilibrium [2]. It is of particular interest for periodically driven systems, which have a discrete time-translation symmetry imposed by the driving. Here, the time symmetry breaking is manifested in the onset of oscillations with a period that is a multiple of the driving period t F . Oscillations with period 2t F due to simultaneously initialized protected boundary states were studied in photonic quantum walks [3]; period-two oscillations can also be expected from the coexistence of Floquet Majorana fermions with quasienergies 0 and π/t F in a cold-atom system [4]. The onset of periodtwo phases was predicted and analyzed [5][6][7][8][9][10] in Floquet many-body localized systems, and the first observations of oscillations at multiples of the driving period in disordered systems were reported [11,12].In systems coupled to a thermal bath, on the other hand, the effect of period doubling has been well-known. A textbook example is a classical oscillator modulated close to twice its eigenfrequency and displaying vibrations with period 2t F [13]. The oscillator has two states of such vibrations; they have opposite phases, reminiscent of a ferromagnet with two orientations of the magnetization. Several aspects of the dynamics of a parametric oscillator in the quantum regime have been studied theoretically, cf. [14][15][16][17][18][19][20][21], and in experiments, cf. [22][23][24]. For a sufficiently strong driving field, a quantum dissipative oscillator, like a classical oscillator, mostly performs vibrations with period 2t F . The interplay of quantum fluctuations and dissipation leads to transitions between the period-two vibrational states, but the rate of these transitions is exponentially small [18].The goal of this paper is to study time symmetry breaking in isolated or almost isolated driven quantum systems with a few degrees of freedom. They are intermediate between large closed systems and dissipative systems, where the nature of the symmetry breaking is very different. To this end, we analyze a driven nonlinear quantum oscillator. Time symmetry breaking in this system should not be limited to period doubling. As an illustration of a behavior qualitatively different from period doubling, we consider period tripling and find the conditions where it occurs. We also address the role of decoherence and the co...

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“…The full distribution of entanglement follows a universal scaling form, and exhibits a bimodal structure that produces universal subleading power-law corrections to the leading volume-law. For systems larger than the correlation length, the short interval entanglement exhibits a discontinuous jump at the transition from fully thermal volume-law on the thermal side, to pure area-law on the MBL side.Recent experimental advances in synthesizing isolated quantum many-body systems, such as cold-atoms [1][2][3][4], trapped ions [5,6], or impurity spins in solids [7,8], have raised fundamental questions about the nature of statistical mechanics. Even when decoupled from external sources of dissipation, large interacting quantum systems tend to act as their own heat-baths and reach thermal equilibrium.…”

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confidence: 99%

Scaling Theory of Entanglement at the Many-Body Localization Transition

2017

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We study the universal properties of eigenstate entanglement entropy across the transition between many-body localized (MBL) and thermal phases. We develop an improved real space renormalization group approach that enables numerical simulation of large system sizes and systematic extrapolation to the infinite system size limit. For systems smaller than the correlation length, the average entanglement follows a sub-thermal volume law, whose coefficient is a universal scaling function. The full distribution of entanglement follows a universal scaling form, and exhibits a bimodal structure that produces universal subleading power-law corrections to the leading volume-law. For systems larger than the correlation length, the short interval entanglement exhibits a discontinuous jump at the transition from fully thermal volume-law on the thermal side, to pure area-law on the MBL side.Recent experimental advances in synthesizing isolated quantum many-body systems, such as cold-atoms [1][2][3][4], trapped ions [5,6], or impurity spins in solids [7,8], have raised fundamental questions about the nature of statistical mechanics. Even when decoupled from external sources of dissipation, large interacting quantum systems tend to act as their own heat-baths and reach thermal equilibrium. This behavior is formalized in the eigenstate thermalization hypothesis (ETH) [9,10]. Generic excited eigenstates of such thermal systems are highly entangled, with the entanglement of a subregion scaling as the volume of that region ("volume law"). This results in incoherent, classical dynamics at long times. In contrast, strong disorder can dramatically alter this picture by pinning excitations that would otherwise propagate heat and entanglement [11][12][13][14][15][16][17]. In such many-body localized (MBL) systems [18][19][20], generic eigenstates have properties akin to those of ground states. They exhibit short-range entanglement that scales like the perimeter of the subregion [17] ("area law"), and have quantum coherent dynamics up to arbitrarily long time scales [21][22][23][24][25][26][27], even at high energy densities [17,26,[28][29][30][31].A transition between MBL and thermal regimes requires a singular rearrangement of eigenstates from area-law to volume-law entanglement. This many-body (de)localization transition (MBLT) represents an entirely new class of critical phenomena, outside the conventional framework of equilibrium thermal or quantum phase transitions. Developing a systematic theory of this transition promises not only to expand our understanding of possible critical phenomena, but also to yield universal insights into the nature of the proximate MBL and thermal phases.The eigenstate entanglement entropy can be viewed as a non-equilibrium analog of the thermodynamic free energy for a conventional thermal phase transition, and plays a central role in our conceptual understanding of the MBL and ETH phases. Describing the entanglement across the MBLT requires addressing the challenging combination of disorder, interactio...

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Observation of discrete time-crystalline order in a disordered dipolar many-body system

Cited by 932 publications

References 40 publications

Connecting dynamical quantum phase transitions and topological steady-state transitions by tuning the energy gap

Connecting dynamical quantum phase transitions and topological steady-state transitions by tuning the energy gap

Time-translation-symmetry breaking in a driven oscillator: From the quantum coherent to the incoherent regime

Scaling Theory of Entanglement at the Many-Body Localization Transition

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