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...

In a magnetic field, electrons in metals repeatedly traverse closed magnetic orbits around the Fermi surface. The resulting oscillations in the density of states enable powerful experimental techniques for measuring a metal's Fermi surface structure. On the other hand, the surface states of Weyl semimetals consist of disjoint, open Fermi arcs raising the question of whether they can be observed by standard quantum oscillatory techniques. Here, we find that the open Fermi arcs participate in unusual closed magnetic orbits by traversing the bulk of the sample to connect opposite surfaces. These orbits have anomalous features that are impossible for conventional surface states, and result in quantum oscillations that contain observable signatures of the topological character of the bulk Weyl semimetal. We also apply our predictions to the compounds Cd 3 As 2 and Na 3 Bi that were recently proposed to be threedimensional Dirac (doubled Weyl) semimetals, and propose experimental signatures of their possible Fermi arc states.

Condensed-matter systems provide a rich setting to realize Dirac and Majorana fermionic excitations as well as the possibility to manipulate them for potential applications. It has recently been proposed that chiral, massless particles known as Weyl fermions can emerge in certain bulk materials or in topological insulator multilayers and give rise to unusual transport properties, such as charge pumping driven by a chiral anomaly. A pair of Weyl fermions protected by crystalline symmetry effectively forming a massless Dirac fermion has been predicted to appear as low-energy excitations in a number of materials termed three-dimensional Dirac semimetals. Here we report scanning tunnelling microscopy measurements at sub-kelvin temperatures and high magnetic fields on the II-V semiconductor Cd3As2. We probe this system down to atomic length scales, and show that defects mostly influence the valence band, consistent with the observation of ultrahigh-mobility carriers in the conduction band. By combining Landau level spectroscopy and quasiparticle interference, we distinguish a large spin-splitting of the conduction band in a magnetic field and its extended Dirac-like dispersion above the expected regime. A model band structure consistent with our experimental findings suggests that for a magnetic field applied along the axis of the Dirac points, Weyl fermions are the low-energy excitations in Cd3As2.

We study the dynamical melting of "hot" one-dimensional many-body localized systems. As disorder is weakened below a critical value, these nonthermal quantum glasses melt via a continuous dynamical phase transition into classical thermal liquids. By accounting for collective resonant tunneling processes, we derive and numerically solve an effective model for such quantum-to-classical transitions and compute their universal critical properties. Notably, the classical thermal liquid exhibits a broad regime of anomalously slow subdiffusive equilibration dynamics and energy transport. The subdiffusive regime is characterized by a continuously evolving dynamical critical exponent that diverges with a universal power at the transition. Our approach elucidates the universal long-distance, low-energy scaling structure of many-body delocalization transitions in one dimension, in a way that is transparently connected to the underlying microscopic physics. We discuss experimentally testable signatures of the predicted scaling properties.

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.