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-

When a closed quantum system is driven periodically with period T , it approaches a periodic state synchronized with the drive in which any local observable measured stroboscopically approaches a steady value. For integrable systems, the resulting behaviour is captured by a periodic version of a generalized Gibbs ensemble. By contrast, here we show that for generic non-integrable interacting systems, local observables become independent of the initial state entirely. Essentially, this happens because Floquet eigenstates of the driven system at quasienergy ωα consist of a mixture of the exponentially many eigenstates of the undriven Hamiltonian which are thus drawn from the entire extensive undriven spectrum. This is a form of equilibration which depends only on the Hilbert space of the undriven system and not on any details of its Hamiltonian.Introduction-There has been intense recent interest in equilibration and thermalization of closed quantum systems. If large enough, such systems approach a steady state welldescribed by the usual constructs of statistical mechanics. The effort to understand the mechanisms by which unitary quantum evolution leads to time-independent states which can be characterised by fixing a reasonably small number of observables, as it must if statistical mechanics is to apply, has been one of the most fruitful in nonequilibrium quantum dynamics [1][2][3][4][5][6][7].At the same time, much experimental and theoretical effort has been devoted to periodically-driven systems [8]. The formal framework has been mostly set up by Shirley [9] and Sambe [10], and has been successfully applied in various fields, such as NMR [11,12], nonlinear optics [13] and others [14][15][16]. Closer to the subject of this work, it has recently been shown that isolated many-body periodically-driven systems eventually synchronize into a periodic steady state with the driving [17,18], in analogy with closed, non-driven systems approaching a stationary equilibrium state.In a recent article [18], we have taken a first step towards characterising the long-time synchronised state, by obtaining a description of the long-time steady-state of an integrable system analogous to the generalized Gibbs ensemble (GGE) [19] for undriven systems, finding that memory of the relevant conserved quantities persists for all time.Here we study the generic situation of a nonintegrable periodically driven model. Remarkably, we find that the longtime behaviour is stationary and independent of both the initial condition and details of the undriven Hamiltonian beyond its Hilbert space.We give a physical mechanism explaining this result: the expectation values of observables in any eigenstate are the same for all eigenstates. This is caused by the width of the quasienergy spectrum being finite, whereas that of the energy spectrum of the undriven Hamiltonian is extensive. This leads to a perturbation theory in the driving having vanishing radius of convergence, instead immediately mixing any initial state with a finite fraction of the states...

We study many-body localized quantum systems subject to periodic driving. We find that the presence of a mobility edge anywhere in the spectrum is enough to lead to delocalization for any driving strength and frequency. By contrast, for a fully localized many-body system, a delocalization transition occurs at a finite driving frequency. We present numerical studies on a system of interacting one-dimensional bosons and the quantum random energy model, as well as simple physical pictures accounting for those results.

The nature of the behaviour of an isolated many-body quantum system periodically driven in time has been an open question since the beginning of quantum mechanics [1][2][3][4][5][6]. After an initial transient, such a system is known to synchronize with the driving; in contrast to the non-driven case, no fundamental principle has been proposed for constructing the resulting non-equilibrium state. Here, we analytically show that, for a class of integrable systems, the relevant ensemble is constructed by maximizing an appropriately defined entropy subject to constraints [7] which we explicitly identify. This result constitutes a generalisation of the concepts of equilibrium statistical mechanics to a class of far-from-equilibrium-systems, up to now mainly accessible using ad-hoc methods.There has recently been significant progress in our understanding of statistical mechanics based on the twin concepts of equilibration, the approach of a large, closed system's state to some steady state [1, 3, 4, 8-10, 12, 13], as well as of thermalization, when this steady state depends only upon a small number of quantities. Starting from ideas due to Jaynes [7], Srednicki and Deutsch [8, 9] and Popescu et al [16], both integrable and non-integrable closed, nondriven many-body systems have thus been shown to thermalize [4, 10, 12].On the other hand, the study of periodically driven systems has also had a long history. Following early foundational work by Shirley [1] and Sambe [2], substantial theoretical and experimental progress has been made recently [3][4][5][6][17][18][19][20][21].Here, we combine ideas from the two areas to extend the concept of thermalization to the out-of-equilibrium case of periodically driven systems. By devising a mapping of the system to a set of effectively non-driven systems we show that a periodically driven system asymptotically approaches a time-periodic steady state at long times (see, e.g., [22] and our Suppl. Mat.). Specializing to a large class of integrable systems, we analytically show that Jaynes' entropy maximisation principle [7] gives a statistical mechanical description of the long-time, synchronized dynamics for infinite systems, and study the approach to this equilibrium state as a function of both the system size and time. Finally, we explain how our proposed setup is achievable with current experimental techniques.Synchronization-The starting point for our analysis is the synchronization of the system with the driving, which may be seen as follows.Consider a time-periodic HamiltonianĤ(t) =Ĥ(t + T ) and denote the time evolution operator over a period starting from time 0 ≤ ε < T byÛ (ε, ε + T ) . Takingh = 1, we define an effective HamiltonianĤ eff viaH eff is a time-independent effective Hamiltonian which takes an initial state at t = 0 to the same final state at t = T as the real time-dependent HamiltonianĤ(t). We concentrate on "stroboscopic" observations, that is, observations at discrete points of time separated by a period, t n = ε + nT for a given ε. The expectation valu...

We propose a non-local interfacial model for 3D short-range wetting at planar and non-planar walls. The model is characterized by a binding potential functional depending only on the bulk Ornstein-Zernike correlation function, which arises from different classes of tube-like fluctuations that connect the interface and the substrate. The theory provides a physical explanation for the origin of the effective position-dependent stiffness and binding potential in approximate local theories, and also obeys the necessary classical wedge covariance relationship between wetting and wedge filling. Renormalization group and computer simulation studies reveal the strong non-perturbative influence of non-locality at critical wetting, throwing light on long-standing theoretical problems regarding the order of the phase transition. [2] are complementary approaches to the theory of confined fluids. Mean-field, non-local density functionals give an accurate description of structural properties but are unable to account correctly for long-wavelength interfacial fluctuations. To understand these it is usually necessary to employ mesoscopic interfacial Hamiltonians based on a collective coordinate l(x), measuring the local interfacial thickness. These models are essentially local in character containing a surface energy term proportional to the stiffness Σ of the unbinding interface and a binding potential function W (l). In more refined theories the stiffness also contains a position dependent term [3], Σ(l), which, it is has been argued, may drive the wetting transition first-order [4]. Despite progress over the last few years there are a number of outstanding problems particularly for wetting with short-ranged forces. In addition, recent studies of fluids in wedge-like geometries have uncovered hidden connections or wedge covariance relations between observables at planar wetting and wedge filling transitions [5], which have yet to understood at a deeper level. In this paper we argue that analogous to developments in density functional methods the general theory of short-ranged three-dimensional wetting should be formulated in terms of a non-local (NL) interfacial Hamiltonian. The model we propose directly allows for bulk-like correlations arising from tube-like fluctuations Consider a Landau-Ginzburg-Wilson Hamiltonian based on a continuum order-parameter (magnetization) m(r) in a semi-infinite geometry with bounding surface described by a single-valued height function ψ(x) where x = (x, y) is the parallel displacement vector. Denoting the surface magnetization by m 1 (x) we write (1) where ds ψ = 1 + (∇ψ) 2 dx is the wall area element whilst φ(m) and φ 1 (m 1 ) are suitable bulk and surface potentials [9]. The bulk Hamiltonian is isotropic so the interfacial tension and stiffness are the same. Following FJ we identify the interfacial modelwhere m Ξ (r) is the profile which minimises Eq. (1) subject to a given interfacial configuration. FJ determined m Ξ (r) perturbatively in terms of local planar constrained profiles [3]. He...