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arXiv:1607.02872 · Physical review. B./Physical review. B

Two-rate periodic protocol with dynamics driven through many cycles

Satyaki Kar

Published Feb 27, 2017• 7 citations
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Abstract

We study the long time dynamics in closed quantum systems periodically driven via time dependent parameters with two frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}=r{\ensuremath{\omega}}_{1}$. Tuning of the ratio $r$ there can unleash plenty of dynamical phenomena to occur. Our study includes integrable models like Ising and $XY$ models in $d=1$ and the Kitaev model in $d=1$ and 2 and can also be extended to Dirac fermions in graphene. We witness the wave-function overlap or dynamic freezing that occurs within some small/ intermediate frequency regimes in the $({\ensuremath{\omega}}_{1},r)$ plane (with $r\ensuremath{\ne}0$) when the ground state is evolved through a single cycle of driving. However, evolved states soon become steady with long driving, and the freezing scenario gets rarer. We extend the formalism of adiabatic-impulse approximation for many cycle driving within our two-rate protocol and show the near-exact comparisons at small frequencies. An extension of the rotating wave approximation is also developed to gather an analytical framework of the dynamics at high frequencies. Finally we compute the entanglement entropy in the stroboscopically evolved states within the gapped phases of the system and observe how it gets tuned with the ratio $r$ in our protocol. The minimally entangled states are found to fall within the regime of dynamical freezing. In general, the results indicate that the entanglement entropy in our driven short-ranged integrable systems follow a genuine nonarea law of scaling and show a convergence (with a $r$ dependent pace) towards volume scaling behavior as the driving is continued for a long time.

DOI
10.1103/physrevb.95.085141

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We study the long time dynamics in closed quantum systems periodically driven via time dependent parameters with two frequencies \ensuremathω1{\ensuremath{\omega}}_{1}\ensuremathω1​ and \ensuremathω2=r\ensuremathω1{\ensuremath{\omega}}_{2}=r{\ensuremath{\omega}}_{1}\ensuremathω2​=r\ensuremathω1​. Tuning of the ratio rrr there can unleash plenty of dynamical phenomena to occur. Our study includes integrable models like Ising and XYXYXY models in d=1d=1d=1 and the Kitaev model in d=1d=1d=1 and 2 and can also be extended to Dirac fermions in graphene. We witness the wave-function overlap or dynamic freezing that occurs within some small/ intermediate frequency regimes in the (\ensuremathω1,r)({\ensuremath{\omega}}_{1},r)(\ensuremathω1​,r) plane (with r\ensuremath≠0r\ensuremath{\ne}0r\ensuremath=0) when the ground state is evolved through a single cycle of driving. However, evolved states soon become steady with long driving, and the freezing scenario gets rarer. We extend the formalism of adiabatic-impulse approximation for many cycle driving within our two-rate protocol and show the near-exact comparisons at small frequencies. An extension of the rotating wave approximation is also developed to gather an analytical framework of the dynamics at high frequencies. Finally we compute the entanglement entropy in the stroboscopically evolved states within the gapped phases of the system and observe how it gets tuned with the ratio rrr in our protocol. The minimally entangled states are found to fall within the regime of dynamical freezing. In general, the results indicate that the entanglement entropy in our driven short-ranged integrable systems follow a genuine nonarea law of scaling and show a convergence (with a rrr dependent pace) towards volume scaling behavior as the driving is continued for a long time.
We study periodically driven closed quantum systems where two parameters of the system Hamiltonian are varied periodically in time with frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}=r{\ensuremath{\omega}}_{1}$. We show that such drives may be used to tune towards dynamics-induced freezing where the wave function of the state of the system after a drive cycle at time $T=2\ensuremath{\pi}/{\ensuremath{\omega}}_{1}$ has almost perfect overlap with the initial state. We locate regions in the $({\ensuremath{\omega}}_{1},r)$ plane where the freezing is near exact for a class of integrable models and a specific nonintegrable model. The integrable models that we study encompass Ising and XY\mathit{XY}XY models in d=1d=1d=1, Kitaev model in d=2d=2d=2, and Dirac fermions in graphene and atop a topological insulator surface, whereas the nonintegrable model studied involves the experimentally realized one-dimensional tilted Bose-Hubbard model in an optical lattice. In addition, we compute the relevant correlation functions of such driven systems and describe their characteristics in the region of the $({\ensuremath{\omega}}_{1},r)$ plane where the freezing is near exact. We supplement our numerical analysis with semianalytic results for integrable driven systems within adiabatic-impulse approximation and discuss experiments which may test our theory.
We show that the correlation functions of a class of periodically driven integrable closed quantum systems approach their steady-state value as ${n}^{\ensuremath{-}(\ensuremath{\alpha}+1)/\ensuremath{\beta}}$, where nnn is the number of drive cycles and $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ denote positive integers. We find that, generically, $\ensuremath{\beta}=2$ within a dynamical phase characterized by a fixed $\ensuremath{\alpha}$; however, its value can change to $\ensuremath{\beta}=3$ or $\ensuremath{\beta}=4$ either at critical drive frequencies separating two dynamical phases or at special points within a phase. We show that such decays are realized in both driven Su-Schrieffer-Heeger (SSH) and one-dimensional (1D) transverse field Ising models, discuss the role of symmetries of the Floquet spectrum in determining $\ensuremath{\beta}$, and chart out the values of $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ realized in these models. We analyze the SSH model for a continuous drive protocol using a Floquet perturbation theory which provides analytical insight into the behavior of the correlation functions in terms of its Floquet Hamiltonian. This is supplemented by an exact numerical study of a similar behavior for the 1D Ising model driven by a square pulse protocol. For both models, we find a crossover timescale nc{n}_{c}nc​ which diverges at the transition. We also unravel a long-time oscillatory behavior of the correlators when the critical drive frequency, ${\ensuremath{\omega}}_{c}$, is approached from below $(\ensuremath{\omega}<{\ensuremath{\omega}}_{c})$. We tie such behavior to the presence of multiple stationary points in the Floquet spectrum of these models and provide an analytic expression for the time period of these oscillations.
A class of integrable models, such as the one-dimensional transverse-field Ising model, responds nonmonotonically to a periodic drive with respect to the driving parameters and freezes almost absolutely for certain combinations of the latter. In this paper, we go beyond the two-band structure of the Ising-like models studied previously and ask whether such an unusual nonmonotonic response and near-absolute freezing occur in integrable systems with a higher number of bands. To this end, we consider a tight-binding model for bilayer graphene subjected to an interlayer potential difference. We find that when the potential is driven periodically, the system responds nonmonotonically to variations in the driving amplitude V0{V}_{0}V0​ and frequency $\ensuremath{\omega}$ and shows near-absolute freezing for certain values of ${V}_{0}/\ensuremath{\omega}$. However, the freezing occurs only in the presence of a constant bias in the driving, i.e., when $V={V}^{\ensuremath{'}}+{V}_{0}cos\ensuremath{\omega}t$. When ${V}^{\ensuremath{'}}=0$, the freezing is switched off for all values of ${V}_{0}/\ensuremath{\omega}$. We support our numerical results with analytical calculations based on a rotating wave approximation. We also give a proposal to realize the driven bilayer system via ultracold atoms in an optical lattice, where the driving can be implemented by shaking the lattice.
Abstract 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 α . Starting from an initial unentangled state, we show that all local quantities synchronize with the driving frequency ω at late times and relax to well-defined steady state values in the thermodynamic limit and after drive cycles for any α and ω . We introduce a distance measure, , that characterizes the approach of the reduced density matrix of a subsystem of l sites to that of the final steady state. We chart out the n dependence of and identify a critical value (which depends only on the time-averaged Hamiltonian) below which they generically decay to zero as . For , in contrast, for with at least one intermediate dynamical transition. An identical behavior is found for relaxation of all non-trivial correlation functions to their steady-state values. We also study the mutual information propagation to understand the nature of the entanglement spreading in space with increasing n for such periodically driven long-ranged systems. We point out existence of qualitatively new features (absent for ) in the space-time dependence of mutual information for , where is the largest critical frequency for the dynamical transition for a given α such as the presence of multiple light cone-like structures which persists even when α is large. We also show that the space-time dependence of the mutual information of long-ranged Hamiltonians with differs qualitatively from those with for any drive frequency and relate this to the behavior of the Floquet group velocity of such driven system.
We study entanglement transitions in a periodically driven Ising chain in the presence of an imaginary transverse field $\ensuremath{\gamma}$ as a function of drive frequency ${\ensuremath{\omega}}_{D}$. In the high drive amplitude and frequency regime, we find a critical value $\ensuremath{\gamma}={\ensuremath{\gamma}}_{c}$ below which the steady-state half-chain entanglement entropy, SL/2{S}_{L/2}SL/2​, scales with chain length LLL as ${S}_{L/2}\ensuremath{\sim}lnL$; in contrast, for $\ensuremath{\gamma}>{\ensuremath{\gamma}}_{c}$, it becomes independent of LLL. In the small $\ensuremath{\gamma}$ limit, we compute the coefficient, $\ensuremath{\alpha}$, of the lnLlnLlnL term analytically using a Floquet perturbation theory and trace its origin to the presence of Fisher-Hartwig jump singularities in the correlation function of the driven chain. We also study the frequency dependence of ${\ensuremath{\gamma}}_{c}$ and show that ${\ensuremath{\gamma}}_{c}\ensuremath{\rightarrow}0$ at special drive frequencies; at these frequencies, which we analytically compute, SL/2{S}_{L/2}SL/2​ remain independent of LLL for all $\ensuremath{\gamma}$. This behavior can be traced to an approximate emergent symmetry of the Floquet Hamiltonian at these drive frequencies which we identify. Finally, we discus the behavior of the driven system at low and intermediate drive frequencies. Our analysis shows the presence of volume law behavior of the entanglement in this regime ${S}_{\ensuremath{\ell}}\ensuremath{\sim}\ensuremath{\ell}$ for small subsystem length $\ensuremath{\ell}\ensuremath{\le}{\ensuremath{\ell}}^{*}({\ensuremath{\omega}}_{D})$. We identify ${\ensuremath{\ell}}^{*}({\ensuremath{\omega}}_{D})$ and tie its existence to the effective long-range nature of the Floquet Hamiltonian of the driven chain for small subsystem size. We discuss the applicability of our results to other integrable non-Hermitian models.
We derive an exact analytical expression, at stroboscopic intervals, for the time-dependent wave function of a class of integrable quantum many-body systems, driven by the periodic delta-kick protocol. To investigate long-time dynamics, we use the wave function to obtain an exact analytical expression for the expectation values of the defect density, magnetization, residual energy, fidelity, and the correlation function after nthn\mathrm{th}nth drive cycle. Periodically driven integrable closed quantum systems absorb energy, and the long-time universal dynamics are described by the periodic generalized Gibbs ensemble (GGE). We demonstrate that the expectation values of all observables are divided into two parts: one highly oscillatory term that depends on the drive cycle nnn, and the rest of the terms are independent of it. Typically, the nnn-independent part constitutes the saturation at large nnn and periodic GGE. The contribution from the highly oscillatory term vanishes in large nnn. We also generalize our formalism to include square-pulse and sinusoidal driving protocols.