In this work we introduce boundary time crystals. Here continuous time-translation symmetry breaking occurs only in a macroscopic fraction of a many-body quantum system. After introducing their definition and properties, we analyze in detail a solvable model where an accurate scaling analysis can be performed. The existence of the boundary time crystals is intimately connected to the emergence of a time-periodic steady state in the thermodynamic limit of a many-body open quantum system. We also discuss connections to quantum synchronization.

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 behaviour. 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 quasi-energies differing of half the driving frequencies, a key essence of Floquet time crystals. Remarkably, the stability of the time-crystal phase can be directly analysed 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.Comment: 14 pages, 12 figure

We study scrambling in connection to multipartite entanglement dynamics in regular and chaotic long-range spin chains, characterized by a well defined semi-classical limit. For regular dynamics, scrambling and entanglement dynamics are found to be very different: up to the Ehrenfest time, they rise side by side departing only afterward. Entanglement saturates and becomes extensively multipartite, while scrambling, characterized by the dynamic of the square commutator of initially commuting variables, continues its growth up to the recurrence time. Remarkably, the exponential growth of the latter emerges not only in the chaotic case but also in the regular one, when the dynamics occurs at a dynamical critical point. III. THE MODEL AND OUT-OF-EQUILIBRIUM PROTOCOLSWe consider an Ising chain in transverse field with longrange interactions,

In this letter we present, in a number conserving framework, a model of interacting fermions in a two-wire geometry supporting non-local zero-energy Majorana-like edge excitations. The model has an exactly solvable line, on varying the density of fermions, described by a topologically non-trivial ground state wave-function. Away from the exactly solvable line we study the system by means of the numerical density matrix renormalization group. We characterize its topological properties through the explicit calculation of a degenerate entanglement spectrum and of the braiding operators which are exponentially localized at the edges. Furthermore, we establish the presence of a gap in its single particle spectrum while the Hamiltonian is gapless, and compute the correlations between the edge modes as well as the superfluid correlations. The topological phase covers a sizeable portion of the phase diagram, the solvable line being one of its boundaries.Introduction -Large part of the enormous attention devoted in the last years to topological superconductors owes to the exotic quasiparticles such as Majorana modes, which localize at their boundaries (edges, vortices, . . . ) [1,2] and play a key role in several robust quantum information protocols [3]. Kitaev's p-wave superconducting quantum wire [4] provides a minimal setting showcasing all the key aspects of topological states of matter in fermionic systems. The existence of a socalled "sweet point" supporting an exact and easy-tohandle analytical solution puts this model at the heart of our understanding of systems supporting Majorana modes. Various implementations in solid state [5,6] and ultracold atoms [7,8] via proximity to superconducting or superfluid reservoirs have been proposed, and experimental signatures of edge modes were reported [9]. Kitaev's model is an effective mean-field model and its Hamiltonian does not commute with the particle number operator. Considerable activity has been devoted to understanding models supporting Majorana edge modes in a number-conserving setting [10][11][12][13][14], as in various experimental platforms (e.g. solid state [10,11] or ultracold atoms [12,13]) this property is naturally present. It was realised that a simple way to promote particle number conservation to a symmetry of the model, while keeping the edge state physics intact, was to consider at least two coupled wires rather than a single one [10][11][12]. However, since attractive interactions are pivotal to generate superconducting order in the canonical ensemble, one usually faces a complex interacting many-body problem. Therefore, approximations such as bosonization [10][11][12], or numerical approaches [13] were invoked. An exactly solvable model of a topological superconductor in a number conserving setting, which would directly complement Kitaev's scenario, is missing (see however [14]).In this letter we present an exactly solvable model of a topological superconductor which supports exotic Majorana-like quasiparticles at its ends and retains the fermionic ...