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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.
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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.
One dimensional systems sometimes show pathologically slow decay of currents. This robustness can be traced to the fact that an integrable model is nearby in parameter space. In integrable models some part of the current can be conserved, explaining this slow decay. Unfortunately, although this conservation law is formally anticipated, in practice it has been difficult to find in concrete cases, such as the Heisenberg model. We investigate this issue both analytically and numerically and find that the appropriate conservation law can be a non-analytic combination of the known local conservation laws and hence is invisible to elementary assumptions.
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 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 are 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.
Driving a quantum system periodically in time can profoundly alter its long-time correlations and give rise to exotic quantum states of matter. The complexity of the combination of many-body correlations and dynamic manipulations has the potential to uncover a whole field of new phenomena, but the theoretical and numerical understanding becomes extremely difficult. We now propose a promising numerical method by generalizing the density matrix renormalization group to a superposition of Fourier components of periodically driven many-body systems using Floquet theory. With this method we can study the full time-dependent quantum solution in a large parameter range for all evolution times, beyond the commonly used high-frequency approximations. Numerical results are presented for the isotropic Heisenberg antiferromagnetic spin-1/2 chain under both local(edge) and global driving for spin-spin correlations and temporal fluctuations. As the frequency is lowered, we demonstrate that more and more Fourier components become relevant and determine strong length- and frequency-dependent changes of the quantum correlations that cannot be described by effective static models.
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