Recent work suggests that a sharp definition of `phase of matter' can be given for periodically driven `Floquet' quantum systems exhibiting many-body localization. In this work we propose a classification of the phases of interacting Floquet localized systems with (completely) spontaneously broken symmetries -- we focus on the one dimensional case, but our results appear to generalize to higher dimensions. We find that the different Floquet phases correspond to elements of $Z(G)$, the centre of the symmetry group in question. In a previous paper we offered a companion classification of unbroken, i.e., paramagnetic phases.
Floquet systems are governed by periodic, time-dependent, Hamiltonians. Prima facie they should absorb energy from the external drives involved in modulating their couplings and heat up to infinite temperature. However this unhappy state of affairs can be avoided in many ways. Instead, as has become clear from much recent work, they can exhibit a variety of nontrivial behavior---some of it impossible in undriven systems. In this review we describe the main ideas and themes of this work: novel Floquet drives which exhibit nontrivial topology in single-particle systems, the existence and classif…
Recent work suggests that a sharp definition of `phase of matter' can be given for some quantum systems out of equilibrium---first for many-body localized systems with time independent Hamiltonians and more recently for periodically driven or Floquet localized systems. In this work we propose a classification of the finite abelian symmetry protected phases of interacting Floquet localized systems in one dimension. We find that the different Floquet phases correspond to elements of $\text{Cl}\times\mathcal{A}_G$, where $\text{Cl}$ is the undriven interacting classification, and $\mathcal{A}_G$ …
We investigate Floquet Topological Insulators in the presence of spatially-modulated light. We extend on previous work to show that light can be used to generate and control localized modes in the bulk of these systems. We provide examples of bulk modes generated through modulation of different properties of the light, such as its phase, polarization and frequency. We show that these effects may be realized in a variety of systems, including a zincblende model and graphene, and also provide a generalization of these results to three dimensional systems.
We construct and classify chiral topological phases in driven (Floquet) systems of strongly interacting bosons, with finite-dimensional site Hilbert spaces, in two spatial dimensions. The construction proceeds by introducing exactly soluble models with chiral edges, which in the presence of many-body localization (MBL) in the bulk are argued to lead to stable chiral phases. These chiral phases do not require any symmetry, and in fact owe their existence to the absence of energy conservation in driven systems. Surprisingly, we show that they are classified by a quantized many-body index, which …
We generalize the recently introduced Density-Matrix Renormalization Group (DMRG-X) [Khemani et al, PRL 2016] algorithm to obtain Floquet eigenstates of one-dimensional, periodically driven many-body localized systems. This generalization is made possible by the fact that the time-evolution operator for a period can be efficiently represented using a matrix-product operator. We first benchmark the method by comparing to exact diagonalization for small systems. We then obtain Floquet eigenstates for larger systems and show unambiguously that the characteristic area-law scaling remains robust.
In this work, we investigate how the critical driving amplitude at the Floquet MBL-to-ergodic phase transition differs between smooth and non-smooth driving over a wide range of driving frequencies. To this end, we study numerically a disordered spin-1/2 chain which is periodically driven by a sine or a square-wave drive, respectively. In both cases, the critical driving amplitude increases monotonically with the frequency, and at large frequencies, it is identical for the two drives in the appropriate normalization. However, at low and intermediate frequencies the critical amplitude of the sq…
We define what it means for time translation symmetry to be spontaneously broken in a quantum system, and show with analytical arguments and numerical simulations that this occurs in a large class of many-body-localized driven systems with discrete time-translation symmetry.
Clean and interacting periodically driven quantum systems are believed to exhibit a single, trivial "infinite-temperature" Floquet-ergodic phase. In contrast, here we show that their disordered Floquet many-body 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.
The fate of many-body localization in long-range interacting systems is not fully settled. For instance, the phase boundary between ergodic and many-body localized regimes is still under debate. Here, we use Floquet dynamics which can induce many-body localization in a clean long-range interacting system through spatiotemporal disorder, which are realized by regular operation of random local rotations. The phase diagram has been determined for two types of uniform and nonuniform long-range couplings. Our Floquet mechanism shows more localizing power than conventional static disorder methods as…
We demonstrate the existence of a two-dimensional anomalous Floquet insulator (AFI) phase: an interacting (periodically-driven) non-equilibrium topological phase of matter with no counterpart in equilibrium. The AFI is characterized by a many-body localized bulk, exhibiting nontrivial micromotion within a driving period, and delocalized (thermalizing) chiral states at its boundaries. For a geometry without edges, we argue analytically that the bulk may be many-body localized in the presence of interactions, deriving conditions where stability is expected. We investigate the interplay between t…
"Floquet engineering" - designing band structures "on-demand" through the application of coherent time-periodic drives - has recently emerged as a powerful tool for creating new topological and anomalous phases of matter. In this manuscript, we show that the same principle can be applied to create non-equilibrium correlated states with spontaneously broken symmetry in a lightly doped semiconductor. The periodic drive provides means for obtaining large electronic densities of states necessary for the broken symmetry phase. The phase transition occurs in the steady-state of the system achieved d…
Periodic driving of a quantum system can enable new topological phases with no analog in static systems. In this paper we systematically classify one-dimensional topological and symmetry-protected topological (SPT) phases in interacting fermionic and bosonic quantum systems subject to periodic driving, which we dub Floquet SPTs (FSPTs). For physical realizations of interacting FSPTs, many-body localization by disorder is a crucial ingredient, required to obtain a stable phase that does not catastrophically heat to infinite temperature. We demonstrate that bosonic and fermionic FSPTs phases are…
We present a method for deriving bulk and edge invariants for interacting, many-body localized Floquet systems in two spatial dimensions. This method is based on a general mathematical object which we call a flow. As an application of our method, we derive bulk invariants for Floquet systems without symmetry, as well as for systems with $U(1)$ symmetry. We also derive new formulations of previously known single-particle and many-body invariants. For bosonic systems without symmetry, our invariant gives a bulk counterpart of the rational-valued GNVW index $\frac{p}{q}$ quantifying transport of …
Symmetry protected topological order in one dimension leads to protected degeneracies between symmetry blocks of the reduced density matrix. In the presence of periodic driving, topological Floquet phases can be identified in terms of a cycling of these symmetry blocks between different charge quantum numbers. We discuss an example of this phenomenon with an Ising $\mathbb{Z}_2$ symmetry, using both analytic methods and real quantum computers. By adiabatically moving along the phase diagram, we demonstrate that the cycling periodicity is broken in Floquet topological phase transitions. An equi…
We study the effects of a periodically driven electric field applied to a variety of tight-binding models in one dimension. We first consider a non-interacting system with or without a staggered on-site potential, and we find that that periodic driving can generate states localized completely or partially near the ends of a finite-sized system. Depending on the system parameters, such states have Floquet eigenvalues lying either outside or inside the continuum of eigenvalues of the bulk states; only in the former case we find that these states are completely localized at the ends and are true …
We extend the notion of fragile topology to periodically-driven systems. We demonstrate driving-induced fragile topology in two different models, namely, the Floquet honeycomb model and the Floquet $π$-flux square-lattice model. In both cases, we discover a rich phase diagram that includes Floquet fragile topological phases protected by crystalline rotation or mirror symmetries, Floquet Chern insulators, and trivial atomic phases, with distinct boundary features. Remarkably, the transitions between different phases can be feasibly achieved by simply tuning the driving amplitudes, which is a un…
We explore how disorder and interactions conspire in lattice models with sequentially activated hopping to produce novel k-body (or many-body) localized phases. Specifically, we show that when disorder is added to the set of interacting floquet models considered in [Wampler and Klich arXiv:2209.09180], regions in parameter space near the special points where classical-like dynamics emerge are stabilized prethermally (or via many-body localization in some cases) producing new families of interesting phases. We also find that this disordered system exhibits novel phases in regions of parameter s…
Detection and manipulation of excitations with non-Abelian statistics, such as Majorana fermions, are essential for creating topological quantum computers. To this end, we show the connection between the existence of such localized particles and the phenomenon of unitary subharmonic response (SR) in periodically driven systems. In particular, starting from highly nonequilibrium initial states, the unpaired Majorana modes exhibit spin oscillations with twice the driving period, are localized, and can have exponentially long lifetimes in clean systems. While the lifetime of SR is limited in tran…
In three-dimensional (3D) quantum magnets, magnonic Weyl points (WPs) featuring linear band crossing of two non-degenerate magnon branches can emerge in certain lattice geometry when time-reversal symmetry is broken macroscopically. Unfortunately, there are very limited 3D quantum magnets that host magnonic WPs, and they are yet to be observed experimentally because the intrinsic perturbative interactions that break time-reversal symmetry macroscopically can be very negligible. Here, we present an alternative means via photo-irradiation, in which magnonic WPs can emerge in 3D quantum magnets w…
Dynamical systems may host a number of remarkable symmetry-protected phases that are qualitatively different from their static analogs. In this work, we consider the phase space of symmetry-respecting unitary evolutions in detail and identify several distinct classes of evolution that host novel dynamical order. Using ideas from group cohomology, we construct a set of interacting Floquet drives that generate dynamical symmetry-protected topological order for each nontrivial cohomology class in every dimension, illustrating our construction with explicit two-dimensional examples. We also identi…
We construct a many-body quantized invariant that sharply distinguishes among two dimensional non-equilibrium driven phases of interacting fermions. This is an interacting generalization of a band-structure Floquet quasi-energy winding number, and describes chiral pumping of quantum information along the edge. In particular, our invariant sharply distinguishes between a trivial and anomalous Floquet Anderson insulator in the interacting, many-body localized setting. It also applies more generally to models where only fermion parity is conserved, where it differentiates between trivial models a…
Universal driving protocol for symmetry-protected Floquet topological phasesWe propose a universal driving protocol for the realization of symmetry-protected topological phases in $2+1$ dimensional Floquet systems. Our proposal is based on the theoretical analysis of the possible symmetries of a square lattice model with pairwise nearest-neighbor coupling terms. Among the eight possible symmetry operators we identify the two relevant choices for topological phases with either time-reversal, chiral, or particle-hole symmetry. From the corresponding symmetry conditions on the protocol parameters…
We consider the spectral statistics of the Floquet operator for disordered, periodically driven spin chains in their quantum chaotic and many-body localized phases (MBL). The spectral statistics are characterized by the traces of powers $t$ of the Floquet operator, and our approach hinges on the fact that, for integer $t$ in systems with local interactions, these traces can be re-expressed in terms of products of dual transfer matrices, each representing a spatial slice of the system. We focus on properties of the dual transfer matrix products as represented by a spectrum of Lyapunov exponents…
We experimentally study a periodically driven many-body localized system realized by interacting fermions in a one-dimensional quasi-disordered optical lattice. By preparing the system in a far-from-equilibrium state and monitoring the remains of an imprinted density pattern, we identify a localized phase at high drive frequencies and an ergodic phase at low ones. These two distinct phases are separated by a dynamical phase transition which depends on both the drive frequency and the drive strength. Our observations are quantitatively supported by numerical simulations and are directly connect…
Periodically driven noninteracting systems may exhibit anomalous chiral edge modes, despite hosting bands with trivial topology. We find that these drives have surprising many-body analogs, corresponding to class A, which exhibit anomalous charge and information transport at the boundary. Drives of this form are applicable to generic systems of bosons, fermions, and spins, and may be characterized by the anomalous unitary operator that acts at the edge of an open system. We find that these operators are robust to all local perturbations and may be classified by a pair of coprime integers. This…
Two-dimensional periodically driven systems can host an unconventional topological phase unattainable for equilibrium systems, termed the Anomalous Floquet-Anderson insulator (AFAI). The AFAI features a quasi-energy spectrum with chiral edge modes and a fully localized bulk, leading to non-adiabatic but quantized charge pumping. Here, we show how such a Floquet phase can be realized in a driven, disordered Quantum Anomalous Hall insulator, which is assumed to have two critical energies where the localization length diverges, carrying states with opposite Chern numbers. Driving the system at a …
Higher-order topological~(HOT) states,~hosting topologically protected modes on lower-dimensional boundaries,~such as hinges and corners, have recently extended the realm of the static topological phases.~Here we demonstrate the possibility of realizing a two-dimensional \emph{Floquet} second-order topological insulator, featuring corner-localized zero quasienergy modes and characterized by quantized Floquet qudrupolar moment $Q^{\rm Flq}_{xy}=0.5$, by periodically kicking a quantum spin Hall insulator (QSHI) with a discrete fourfold ($C_4$) and time-reversal (${\mathcal T}$) symmetry breaking…
