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...

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. Spontaneous symmetry breaking-where a quantum state breaks an underlying symmetry of its parent Hamiltonian-represents a unifying concept in modern physics [1, 2]. Its ubiquity spans from condensed matter and atomic physics to high energy particle physics; indeed, examples of the phenomenon abound in nature: superconductors, Bose-Einstein condensates, (anti)-ferromagnets, any crystal, and Higgs mass generation for fundamental particles. This diversity seems to suggest that almost any symmetry can be broken.Spurred by this notion, and the analogy to spatial crystals, Wilczek proposed the intriguing concept of a "timecrystal"-a state which spontaneously breaks continuous time translation symmetry [3][4][5]. Subsequent work developed more precise definitions of such time translation symmetry breaking (TTSB) [6][7][8] and ultimately led to a proof of the "absence of (equilibrium) quantum time crystals" [9]. However, this proof leaves the door open to TTSB in an intrinsically out-of-equilibrium setting, and pioneering recent work [10,11] has demonstrated that quantum systems subject to periodic driving can indeed exhibit discrete TTSB [10-13]; such systems develop persistent macroscopic oscillations at an integer multiple of the driving period, manifesting in a sub-harmonic response for physical observables.An important constraint on symmetry breaking in many-body Floquet systems is the need for disorder and localization [10][11][12][13][14][15][16][17][18]. In the translation-invariant setting, Floquet eigenstates are short-range correlated and resemble infinite temperature states which cannot exhibit symmetry breaking [16,19,20]. Under certain conditions, however, prethermal time-crystal-like dynamics can persist for long times [21,22] even in the absence of localization before ultimately being destroyed by thermalization [18,23].In this Letter, we present three main results.

This corrects the article DOI: 10.1103/PhysRevLett.118.030401.

We propose and analyze two distinct routes toward realizing interacting symmetry-protected topological (SPT) phases via periodic driving. First, we demonstrate that a driven transversefield Ising model can be used to engineer complex interactions which enable the emulation of an equilibrium SPT phase. This phase remains stable only within a parametric time scale controlled by the driving frequency, beyond which its topological features break down. To overcome this issue, we consider an alternate route based upon realizing an intrinsically Floquet SPT phase that does not have any equilibrium analog. In both cases, we show that disorder, leading to many-body localization, prevents runaway heating and enables the observation of coherent quantum dynamics at high energy densities. Furthermore, we clarify the distinction between the equilibrium and Floquet SPT phases by identifying a unique micromotion-based entanglement spectrum signature of the latter. Finally, we propose a unifying implementation in a one-dimensional chain of Rydbergdressed atoms and show that protected edge modes are observable on realistic experimental time scales.The discovery of topological insulators-materials which are insulating in their interior but can conduct on their surface-has led to a multitude of advances at the interface of condensed matter physics and materials engineering [1][2][3][4][5]. At their core, such insulators are characterized by the existence of nontrivial topology in their underlying single-particle electronic band structure [6, 7]. Generalizing our understanding of topological phases to the presence of strong many-body interactions represents one of the central questions in modern physics. Some of the simplest generalizations that have emerged along this direction are symmetry-protected topological (SPT) phases [8][9][10], which represent the minimal extension of topological band insulators to include many-body correlations. Featuring short-range entanglement, SPT phases do not exhibit anyonic excitations in their bulk, but nevertheless possess protected edge modes on their surface; as a result, they represent a particularly fertile ground for studying the interplay between symmetry, topology, and interactions.While indirect signatures of certain ground state SPTs have been observed in the solid state [11][12][13], directly probing the quantum coherence of their underlying edge modes represents an outstanding experimental challenge. In principle, cold-atom quantum simulations could offer a powerful additional tool set-including locally-resolved measurements and interferometric protocols-for probing the robustness of edge modes and systematically exploring their stability to specific perturbations [14][15][16][17][18]. Moreover, such platforms could also enable the controlled storage and transmission of quantum information [19][20][21]. Despite these advantages, and owing to the complexity of typical model SPT Hamiltonians, it remains difficult to engineer and stabilize SPT phases in coldatom systems. One approach to t...

Recent work by De Roeck et al. [Phys. Rev. B 95, 155129 (2017)] has argued that many-body localization (MBL) is unstable in two and higher dimensions due to a thermalization avalanche triggered by rare regions of weak disorder. To examine these arguments, we construct several models of a finite ergodic bubble coupled to an Anderson insulator of non-interacting fermions. We first describe the ergodic region using a GOE random matrix and perform an exact diagonalization study of small systems. The results are in excellent agreement with a refined theory of the thermalization avalanche that includes transient finite size effects, lending strong support to the avalanche scenario. We then explore the limit of large system sizes by modeling the ergodic region via a Hubbard model with all-to-all random hopping-the combined system, consisting of the bubble and the insulator, can be reduced to an effective Anderson impurity problem. We find that the spectral function of a local operator in the ergodic region changes dramatically when it is coupled to a large number of localized fermionic states. This violates a central assumption in the arguments of De Roeck et al. and it may lead to the failure of the avalanche for a given size of the ergodic bubble. However, this back-action effect is suppressed and the avalanche can be recovered if the ergodic bubble is large enough. Thus, the main effect of the back-action is to renormalize the critical bubble size.