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Register a new userEntanglement generation in periodically driven integrable systems: dynamical phase transitions and steady state
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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.
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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 behavior 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 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 state, 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.
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 prototypical 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 examine skyrmions driven periodically over random quenched disorder and show that there is a transition from reversible motion to a state in which the skyrmion trajectories are chaotic or irreversible. We find that the characteristic time required for the system to organize into a steady reversible or irreversible state exhibits a power law divergence near a critical ac drive period, with the same exponent as that observed for reversible to irreversible transitions in periodically sheared colloidal systems, suggesting that the transition can be described as an absorbing phase transition in the directed percolation universality class. We compare our results to the behavior of an overdamped system and show that the Magnus term enhances the irreversible behavior by increasing the number of dynamically accessible orbits. We discuss the implications of this work for skyrmion applications involving the long time repeatable dynamics of dense skyrmion arrays.
Random dynamical systems (RDS) evolve by a dynamical rule chosen independently with a certain probability, from a given set of deterministic rules. These dynamical systems in an interval reach a steady state with a unique well-defined probability density only under certain conditions, namely Pelikans criterion. We investigate and characterize the steady state of a bounded RDS when Pelikans criterion breaks down. In this regime, the system is attracted to a common fixed point (CFP) of all the maps, which is attractive for at least one of the constituent mapping functions. If there are many such fixed points, the initial density is shared among the CFPs; we provide a mapping of this problem with the well known hitting problem of random walks and find the relative weights at different CFPs. The weights depend upon the initial distribution.
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