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Dynamics of Entanglement generation in periodically driven integrable systems

Arnab Sen, K. Sengupta

Published Nov 11, 2015• 0 citations
OpenAlex

Abstract

We study entanglement entropy $S_n$ of periodically driven $d-$dimensional integrable models after $n$ drive cycles with frequency $\omega$. We demonstrate that such a drive may be used for controlled generation of states with non-area-law scaling of $S_n$ and provide a criteria for their occurrence which constitutes a generalization of Hastings' theorem to driven integrable systems. We find that $S_n$ decays to $S_{\infty}$ as $(\omega/n)^{(d+2)/2}$ for fast and $(\omega/n)^{d/2}$ for slow drives; these two dynamical phases are separated by a transition associated with the change in topology of the spectrum of the system's Floquet Hamiltonian. We show that these dynamical phases show re-entrant behavior as a function of $\omega$ for $d=1$ (and a class of $d=2$) models and discuss experiments which can test our theory.

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We study entanglement entropy SnS_nSn​ of periodically driven d−d-d−dimensional integrable models after nnn drive cycles with frequency ω\omegaω. We demonstrate that such a drive may be used for controlled generation of states with non-area-law scaling of SnS_nSn​ and provide a criteria for their occurrence which constitutes a generalization of Hastings' theorem to driven integrable systems. We find that SnS_nSn​ decays to S∞S_{\infty}S∞​ as (ω/n)(d+2)/2(\omega/n)^{(d+2)/2}(ω/n)(d+2)/2 for fast and (ω/n)d/2(\omega/n)^{d/2}(ω/n)d/2 for slow drives; these two dynamical phases are separated by a transition associated with the change in topology of the spectrum of the system's Floquet Hamiltonian. We show that these dynamical phases show re-entrant behavior as a function of ω\omegaω for d=1d=1d=1 (and a class of d=2d=2d=2) models and discuss experiments which can test our theory.
We study a class of periodically driven ddd-dimensional integrable models and show that after nnn drive cycles with frequency $\ensuremath{\omega}$, pure states with non-area-law entanglement entropy ${S}_{n}(l)\ensuremath{\sim}{l}^{\ensuremath{\alpha}(n,\ensuremath{\omega})}$ are generated, where lll is the linear dimension of the subsystem, and $d\ensuremath{-}1\ensuremath{\le}\ensuremath{\alpha}(n,\ensuremath{\omega})\ensuremath{\le}d$. The exponent $\ensuremath{\alpha}(n,\ensuremath{\omega})$ eventually approaches ddd (volume law) for large enough lll when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We identify and analyze the crossover phenomenon from an area $(S\ensuremath{\sim}{l}^{d\ensuremath{-}1}$ for $d\ensuremath{\ge}1$) to a volume $(S\ensuremath{\sim}{l}^{d})$ law and provide a criterion for their occurrence which constitutes a generalization of Hastings's theorem to driven integrable systems in one dimension. We also find that Sn{S}_{n}Sn​ generically decays to ${S}_{\ensuremath{\infty}}$ as ${(\ensuremath{\omega}/n)}^{(d+2)/2}$ for fast and ${(\ensuremath{\omega}/n)}^{d/2}$ for slow periodic drives; these two dynamical phases are separated by a topological transition in the eigenspectrum 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 $\ensuremath{\omega}$ for d=1d=1d=1 models and also discuss the dynamical transition for d>1d>1d>1 models. Finally, we study entanglement properties of the steady state and show that singular features (cusps and kinks in d=1d=1d=1) appear in ${S}_{\ensuremath{\infty}}$ as a function of $\ensuremath{\omega}$ whenever there is a crossing of the Floquet bands. We discuss experiments which can test our theory.
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 study the dynamics of entanglement asymmetry in periodically driven quantum systems. Using a periodically driven XY chain as a model for a driven integrable quantum system, we provide semi-analytic results for the behavior of the dynamics of the entanglement asymmetry, \Delta S Δ S , as a function of the drive frequency. Our analysis identifies special drive frequencies at which the driven XY chain exhibits dynamic symmetry restoration and displays quantum Mpemba effect over a long timescale; we identify an emergent approximate symmetry in its Floquet Hamiltonian which plays a crucial role for realization of both these phenomena. We follow these results by numerical computation of \Delta S Δ S for the non-integrable driven Rydberg atom chain and obtain similar emergent-symmetry-induced symmetry restoration and quantum Mpemba effect in the prethermal regime for such a system. Finally, we provide an exact analytic computation of the entanglement asymmetry for a periodically driven conformal field theory (CFT) on a strip. Such a driven CFT, depending on the drive amplitude and frequency, exhibits two distinct phases, heating and non-heating, that are separated by a critical line. Our results show that for m m cycles of a periodic drive with time period T T , \Delta S \sim \ln mT Δ S ∼ ln m T [ \ln (\ln mT) ln ( ln m T ) ] in the heating phase [on the critical line] for a generic CFT; in contrast, in the non-heating phase, \Delta S Δ S displays small amplitude oscillations around it’s initial value as a function of mT m T . We provide a phase diagram for the behavior of \Delta S Δ S for such driven CFTs as a function of the drive frequency and amplitude.
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.
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 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 $({\ensuremath{\omega}}_{1},r)$ 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 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 the dynamics of a class of integrable non-Hermitian free-fermionic models driven periodically using a continuous drive protocol characterized by an amplitude g1{g}_{1}g1​ and frequency ${\ensuremath{\omega}}_{D}$. We derive an analytic, albeit perturbative, Floquet Hamiltonian for describing such systems using Floquet perturbation theory with ${g}_{1}^{\ensuremath{-}1}$ being the perturbation parameter. Our analysis indicates the existence of special drive frequencies at which an approximately conserved quantity emerges. The presence of such an almost conserved quantity is reflected in the dynamics of the fidelity, the correlation functions, and the half-chain entanglement entropy of the driven system. In addition, it also controls the nature of the steady state of the system. We show that one-dimensional transverse field Ising model, with an imaginary component of the transverse field, serves as an experimentally relevant example of this phenomenon. In this case, the transverse magnetization is approximately conserved; this conservation leads to complete suppression of oscillatory features in the transient dynamics of fidelity, magnetization, and entanglement of the driven chain at special drive frequencies. We discuss the nature of the steady state of the Ising chain near and away from these special frequencies, demonstrate the protocol independence of this phenomenon by showing its existence for discrete drive protocols, and suggest experiments which can test our theory.