Clean and interacting periodically-driven systems are believed to exhibit a single, trivial "infinitetemperature" Floquet-ergodic phase. In contrast, here we show that their disordered Floquet manybody localized counterparts can exhibit distinct ordered phases delineated by sharp transitions. Some of these are analogs of equilibrium states with broken symmetries and topological order, while others -genuinely new to the Floquet problem -are characterized by order and non-trivial periodic dynamics. We illustrate these ideas in driven spin chains with Ising symmetry.Introduction: Extending ideas from equilibrium statistical mechanics to the non-equilibrium setting is a topic of perennial interest. We consider a question in this vein: Is there a sharp notion of a phase in driven, interacting quantum systems? We find an affirmative answer for Floquet systems 1-3 whose Hamiltonians depend on time t periodically, H(t + T ) = H(t). Unlike in equilibrium statistical mechanics, disorder turns out to be an essential ingredient for stabilizing different phases; moreover, the periodic time evolution allows for the existence (and diagnosis) of phases without any counterparts in equilibrium statistical mechanics.Naively, Floquet systems hold little promise of a complex phase structure. In systems with periodic Hamiltonians, not even the basic concept of energy survives, being replaced instead with a quasi-energy defined up to arbitrary shifts of 2π/T . Indeed, interacting Floquet systems should absorb energy indefinitely from the driving field, as suggested by standard linear response reasoning wherein any nonzero frequency exhibits dissipation. This results in the system heating up to "infinite temperature", at which point all static and dynamic correlations become trivial and independent of starting state -thus exhibiting a maximally trivial form of ergodicity 4-6 . To get anything else requires a mechanism for energy localization wherein the absorption from the driving field saturates, and the long-time state of the system is sensitive to initial conditions. The current dominant belief is that translationally invariant interacting systems cannot generically exhibit such energy localization 4-6 , although there are computations that suggest otherwise 7-9 . The basic intuition is that spatially extended modes in translationally invariant systems interact with and transfer energy between each other.This can be different when disorder spatially localizes the modes, with individual modes exhibiting something like Rabi oscillations while interacting only weakly with distant modes. While the actual situation is somewhat more involved, several pieces of work 10-12 have made a convincing case for the existence of Floquet energy localization exhibiting a set of properties closely related to those exhibited by time-independent many-body localized 13 (MBL) systems 14 . In the following we show that such Floquet-MBL sys-

A fundamental assumption in statistical physics is that generic closed quantum many-body systems thermalize under their own dynamics. Recently, the emergence of many-body localized systems has questioned this concept and challenged our understanding of the connection between statistical physics and quantum mechanics. Here we report on the observation of a many-body localization transition between thermal and localized phases for bosons in a two-dimensional disordered optical lattice. With our single-site-resolved measurements, we track the relaxation dynamics of an initially prepared out-of-equilibrium density pattern and find strong evidence for a diverging length scale when approaching the localization transition. Our experiments represent a demonstration and in-depth characterization of many-body localization in a regime not accessible with state-of-the-art simulations on classical computers.

Understanding quantum dynamics away from equilibrium is an outstanding challenge in the modern physical sciences. It is well known that out-of-equilibrium systems can display a rich array of phenomena, ranging from self-organized synchronization to dynamical phase transitions1,2. More recently, advances in the controlled manipulation of isolated many-body systems have enabled detailed studies of non-equilibrium phases in strongly interacting quantum matter3-6. As a particularly striking example, the interplay of periodic driving, disorder, and strong interactions has recently been predicted to result in exotic "time-crystalline" phases7, which spontaneously break the discrete time-translation symmetry of the underlying drive8-11. Here, we report the experimental observation of such discrete time-crystalline order in a driven, disordered †

We study the scrambling of local quantum information in chaotic many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively. The interplay between conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a slow dissipative process. We obtain our results in a random quantum circuit model that is constrained to have a conservation law. We find that a generic spreading operator consists of two parts: (i) a conserved part which comprises the weight of the spreading operator on the local conserved densities, whose dynamics is described by diffusive charge spreading. This conserved part also acts as a source that steadily emits a flux of (ii) non-conserved operators. This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the slow conversion of operator weight from local conserved operators to nonconserved, at a rate set by the local diffusion current. The emitted nonconserved parts then spread ballistically at a butterfly speed, thus becoming highly nonlocal and hence essentially non-observable, thereby acting as the "reservoir" that facilitates the dissipation. In addition, we find that the nonconserved component develops a power law tail behind its leading ballistic front due to the slow dynamics of the conserved components. This implies that the out-of-time-order commutator (OTOC) between two initially separated operators grows sharply upon the arrival of the ballistic front but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially. We also derive these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.