We study the many-body localization (MBL) transition of Floquet eigenstates in a driven, interacting fermionic chain with an incommensurate Aubry-Andr{e} potential and a time-periodic hopping amplitude as a function of the drive frequency $omega_D$ u
sing exact diagonalization (ED). We find that the nature of the Floquet eigenstates change from ergodic to Floquet-MBL with increasing frequency; moreover, for a significant range of intermediate $omega_D$, the Floquet eigenstates exhibit non-trivial fractal dimensions. We find a possible transition from the ergodic to this multifractal phase followed by a gradual crossover to the MBL phase as the drive frequency is increased. We also study the fermion auto-correlation function, entanglement entropy, normalized participation ratio (NPR), fermion transport and the inverse participation ratio (IPR) as a function of $omega_D$. We show that the auto-correlation, fermion transport and NPR displays qualitatively different characteristics (compared to their behavior in the ergodic and MBL regions) for the range of $omega_D$ which supports multifractal eigenstates. In contrast, the entanglement growth in this frequency range tend to have similar features as in the MBL regime; its rate of growth is controlled by $omega_D$. Our analysis thus indicates that the multifractal nature of Floquet-MBL eigenstates can be detected by studying auto-correlation function and fermionic transport of these driven chains. We support our numerical results with a semi-analytic expression of the Floquet Hamiltonian obtained using Floquet perturbation theory (FPT) and discuss possible experiments which can test our predictions.
We present a brief overview of some of the analytic perturbative techniques for the computation of the Floquet Hamiltonian for a periodically driven, or Floquet, quantum many-body system. The key technical points about each of the methods discussed a
re presented in a pedagogical manner. They are followed by a brief account of some chosen phenomena where these methods have provided useful insights. We provide an extensive discussion of the Floquet-Magnus expansion, the adiabatic-impulse approximation, and the Floquet perturbation theory. This is followed by a relatively short discourse on the rotating wave approximation, a Floquet-Magnus resummation technique and the Hamiltonian flow method. We also provide a discussion of some open problems which may possibly be addressed using these methods.
We study the dynamics of the periodically driven Rydberg chain starting from the state with zero Rydberg excitations (vacuum state denoted by $|0rangle$) using a square pulse protocol in the high drive amplitude limit. We show, using exact diagonaliz
ation for finite system sizes ($Lle 26$), that the Floquet Hamiltonian of the system, within a range of drive frequencies which we chart out, hosts a set of quantum scars which have large overlap with the $|0rangle$ state. These scars are distinct from their counterparts having high overlap with the maximal Rydberg excitation state ($|mathbb{Z}_2rangle$); they coexist with the latter class of scars and lead to persistent coherent oscillations of the density-density correlator starting from the $|0rangle$ state. We also identify special drive frequencies at which the system undergoes perfect dynamic freezing and provide an analytic explanation for this phenomenon. Finally, we demonstrate that for a wide range of drive frequencies, the system reaches a steady state with sub-thermal values of the density-density correlator. The presence of such sub-thermal steady states, which are absent for dynamics starting from the $|mathbb{Z}_2rangle$ state, imply a weak violation of the eigenstate thermalization hypothesis in finite sized Rydberg chains distinct from that due to the scar-induced persistent oscillations reported earlier. We conjecture that in the thermodynamic limit such states may exist as pre-thermal steady states that show anomalously slow relaxation. We supplement our numerical results by deriving an analytic expression for the Floquet Hamiltonian using a Floquet perturbation theory in the high amplitude limit which provides an analytic, albeit qualitative, understanding of these phenomena at arbitrary drive frequencies. We discuss experiments which can test our theory.
We study the unitary dynamics of randomly or quasi-periodically driven tilted Bose-Hubbard (tBH) model in one dimension deep inside its Mott phase starting from a $mathbb{Z}_2$ symmetry-broken state. The randomness is implemented via a telegraph nois
e protocol in the drive period while the quasi-periodic drive is chosen to correspond to a Thue-Morse sequence. The periodically driven tBH model (with a square pulse protocol characterized by a time period $T$) is known to exhibit transitions from dynamical regimes with long-time coherent oscillations to those with rapid thermalization. Here we show that starting from a regime where the periodic drive leads to rapid thermalization, a random drive, which consists of a random sequence of square pulses with period $T+alpha dT$, where $alpha=pm 1$ is a random number and $dT$ is the amplitude of the noise, restores long-time coherent oscillations for special values of $dT$. A similar phenomenon can be seen for a quasi-periodic drive following a Thue-Morse sequence where such coherent behavior is shown to occur for a larger number of points in the $(T, dT)$ plane due to the additional structure of the drive protocol. We chart out the dynamics of the system in the presence of such aperiodic drives, provide a qualitative analytical understanding of this phenomenon, point out the role of quantum scars behind it, and discuss experiments which can test our theory.
The presence of quantum scars, athermal eigenstates of a many-body Hamiltonian with finite energy density, leads to absence of ergodicity and long-time coherent dynamics in closed quantum systems starting from simple initial states. Such non-ergodic
coherent dynamics, where the system does not explore its entire phase space, has been experimentally observed in a chain of ultracold Rydberg atoms. We show, via study of a periodically driven Rydberg chain, that the drive frequency acts as a tuning parameter for several reentrant transitions between ergodic and non-ergodic regimes. The former regime shows rapid thermalization of correlation functions and absence of scars in the spectrum of the systems Floquet Hamiltonian. The latter regime, in contrast, has scars in its Floquet spectrum which control the long-time coherent dynamics of correlation functions. Our results open a new possibility of drive frequency-induced tuning between ergodic and non-ergodic dynamics in experimentally realizable disorder-free quantum many-body systems.
Driven many-body quantum systems where some parameter in the Hamiltonian is varied quasiperiodically in time may exhibit nonequilibrium steady states that are qualitatively different from their periodically driven counterparts. Here we consider a pro
totypical integrable spin system, the spin-$1/2$ transverse field Ising model in one dimension, in a pulsed magnetic field. The time dependence of the field is taken to be quasiperiodic by choosing the pulses to be of two types that alternate according to a Fibonacci sequence. We show that a novel steady state emerges after an exponentially long time when local properties (or equivalently, reduced density matrices of subsystems with size much smaller than the full system) are considered. We use the temporal evolution of certain coarse-grained quantities in momentum space to understand this nonequilibrium steady state in more detail and show that unlike the previously known cases, this steady state is neither described by a periodic generalized Gibbs ensemble nor by an infinite temperature ensemble. Finally, we study a toy problem with a single two-level system driven by a Fibonacci sequence; this problem shows how sensitive the nature of the final steady state is to the different parameters.
We study ramp and periodic dynamics of ultracold bosons in an one-dimensional (1D) optical lattice which supports quantum critical points separating a uniform and a $Z_3$ or $Z_4$ symmetry broken density-wave ground state. Our protocol involves both
linear and periodic drives which takes the system from the uniform state to the quantum critical point (for linear drive protocol) or to the ordered state and back (for periodic drive protocols) via controlled variation of a parameter of the system Hamiltonian. We provide exact numerical computation, for finite-size boson chains with $L le 24$ using exact-diagonalization (ED), of the excitation density $D$, the wavefunction overlap $F$, and the excess energy $Q$ at the end of the drive protocol. For the linear ramp protocol, we identify the range of ramp speeds for which $D$ and $Q$ shows Kibble-Zurek scaling. We find, based on numerical analysis with $L le 24$, that such scaling is consistent with that expected from critical exponents of the $q$-state Potts universality class with $q=3,4$. For periodic protocol, we show that the model display near-perfect dynamical freezing at specific frequencies; at these frequencies $D, Q to 0$ and $|F| to 1$. We provide a semi-analytic explanation of such freezing behavior and relate this phenomenon to a many-body version of Stuckelberg interference. We suggest experiments which can test our theory.
We study periodically driven closed systems with a long-ranged Hamiltonian by considering a generalized Kitaev chain with pairing terms which decay with distance as a power law characterized by exponent $alpha$. Starting from an initial unentangled s
tate, we show that all local quantities relax to well-defined steady state values in the thermodynamic limit and after $n gg 1$ drive cycles for any $alpha$ and driving frequency $omega$. We introduce a distance measure, $mathcal{D}_l(n)$, that characterizes the approach of the reduced density matrix of a subsystem of $l$ sites to its final steady state. We chart out the $n$ dependence of ${mathcal D}_l(n)$ and identify a critical value $alpha=alpha_c$ below which they generically decay to zero as $(omega/n)^{1/2}$. For $alpha > alpha_c$, in contrast, ${mathcal D}_l(n) sim (omega/n)^{3/2}[(omega/n)^{1/2}]$ for $omega to infty [0]$ with at least one intermediate dynamical transition. We also study the mutual information propagation to understand the nature of the entanglement spreading in space with increasing $n$ for such systems. We point out existence of qualitatively new features in the space-time dependence of mutual information for $omega < omega^{(1)}_c$, where $omega^{(1)}_c$ is the largest critical frequency for the dynamical transition for a given $alpha$. One such feature is the presence of {it multiple} light cone-like structures which persists even when $alpha$ is large. We also show that the nature of space-time dependence of the mutual information of long-ranged Hamiltonians with $alpha le 2$ differs qualitatively from their short-ranged counterparts with $alpha > 2$ for any drive frequency and relate this difference to the behavior of the Floquet group velocity of such driven system.
Does a closed quantum many-body system that is continually driven with a time-dependent Hamiltonian finally reach a steady state? This question has only recently been answered for driving protocols that are periodic in time, where the long time behav
ior of the local properties synchronize with the drive and can be described by an appropriate periodic ensemble. Here, we explore the consequences of breaking the time-periodic structure of the drive with additional aperiodic noise in a class of integrable systems. We show that the resulting unitary dynamics leads to new emergent steady states in at least two cases. While any typical realization of random noise causes eventual heating to an infinite temperature ensemble for all local properties in spite of the system being integrable, noise which is self-similar in time leads to an entirely different steady state, which we dub as geometric generalized Gibbs ensemble, that emerges only after an astronomically large time scale. To understand the approach to steady state, we study the temporal behavior of certain coarse-grained quantities in momentum space that fully determine the reduced density matrix for a subsystem with size much smaller than the total system. Such quantities provide a concise description for any drive protocol in integrable systems that are reducible to a free fermion representation.
We study a class of periodically driven $d-$dimensional integrable models and show that after $n$ drive cycles with frequency $omega$, pure states with non-area-law entanglement entropy $S_n(l) sim l^{alpha(n,omega)}$ are generated, where $l$ is the
linear dimension of the subsystem, and $d-1 le alpha(n,omega) le d$. We identify and analyze the crossover phenomenon from an area ($S sim l^{ d-1}$ for $dgeq1$) to a volume ($S sim l^{d}$) law and provide a criterion for their occurrence which constitutes a generalization of Hastings theorem to driven integrable systems in one dimension. We also find that $S_n$ generically decays to $S_{infty}$ as $(omega/n)^{(d+2)/2}$ for fast and $(omega/n)^{d/2}$ for slow periodic drives; these two dynamical phases are separated by a topological transition in the eigensprectrum of the Floquet Hamiltonian. This dynamical transition manifests itself in the temporal behavior of all local correlation functions and does not require a critical point crossing during the drive. We find that these dynamical phases show a rich re-entrant behavior as a function of $omega$ for $d=1$ models, and also discuss the dynamical transition for $d>1$ models. Finally, we study entanglement properties of the steady state and show that singular features (cusps and kinks in $d=1$) appear in $S_{infty}$ as a function of $omega$ whenever there is a crossing of the Floquet bands. We discuss experiments which can test our theory.
