The out-of-equilibrium dynamics of many body systems has recently received a burst of interest, also due to experimental implementations. The dynamics of observables, such as magnetization and susceptibilities, and quantum information related quantities, such as concurrence and entanglement entropy, have been investigated under different protocols bringing the system out of equilibrium. In this paper we focus on the entanglement entropy dynamics under a sinusoidal drive of the tranverse magnetic field in the 1D quantum Ising model. We find that the area and the volume law of the entanglement entropy coexist under periodic drive for an initial non-critical ground state. Furthermore, starting from a critical ground state, the entanglement entropy exhibits finite size scaling even under such a periodic drive. This critical-like behaviour of the out-of-equilibrium driven state can persist for arbitrarily long time, provided that the entanglement entropy is evaluated on increasingly subsytem sizes, whereas for smaller sizes a volume law holds. Finally, we give an interpretation of the simultaneous occurrence of critical and non-critical behaviour in terms of the propagation of Floquet quasi-particles.
Universality is key to the theory of phase transitions, stating that the equilibrium properties of observables near a phase transition can be classified according to few critical exponents. These exponents rule an universal scaling behaviour that witnesses the irrelevance of the model's microscopic details at criticality. Here we discuss the persistence of such a scaling in a one-dimensional quantum Ising model under sinusoidal modulation in time of its transverse magnetic field. We show that scaling of various quantities (concurrence, entanglement entropy, magnetic and fidelity susceptibility) endures up to a stroboscopic time tbd, proportional to the size of the system. This behaviour is explained by noticing that the low-energy modes, responsible for the scaling properties, are resilient to the absorption of energy. Our results suggest that relevant features of the universality do hold also when the system is brought out-of-equilibrium by a periodic driving.
Abstract Universality is key to the theory of phase transitions, stating that the equilibrium properties of observables near a phase transition can be classified according to few critical exponents. These exponents rule an universal scaling behaviour that witnesses the irrelevance of the model’s microscopic details at criticality. Here we discuss the persistence of such a scaling in a one-dimensional quantum Ising model under sinusoidal modulation in time of its transverse magnetic field. We show that scaling of various quantities (concurrence, entanglement entropy, magnetic and fidelity susceptibility) endures up to a stroboscopic time τ bd , proportional to the size of the system. This behaviour is explained by noticing that the low-energy modes, responsible for the scaling properties, are resilient to the absorption of energy. Our results suggest that relevant features of the universality do hold also when the system is brought out-of-equilibrium by a periodic driving.
Abstract In this work we study the entanglement entropy of a uniform quantum Ising chain in transverse field undergoing a periodic driving of period τ . By means of Floquet theory we show that, for any subchain, the entanglement entropy tends asymptotically to a value τ -periodic in time. We provide a semi-analytical formula for the leading term of this asymptotic regime: It is constant in time and obeys a volume law. The entropy in the asymptotic regime is always smaller than the thermal one: because of integrability the system locally relaxes to a generalized Gibbs ensemble (GGE) density matrix. The leading term of the asymptotic entanglement entropy is completely determined by this GGE density matrix. Remarkably, the asymptotic entropy shows marked features in correspondence to some non-equilibrium quantum phase transitions undergone by a Floquet state analog of the ground state.
We numerically study the dynamics of entanglement entropy, induced by an oscillating time periodic driving of the transverse field,
h(t), of a one-dimensional quantum Ising ring. We consider several realizations of
h(t), and we find a number of results in analogy with entanglement entropy dynamics induced by a sudden quantum quench. After a short-time relaxation, the dynamics of entanglement entropy synchronizes with
h(t), displaying an oscillatory behavior at the frequency of the driving. Synchronization in the dynamics of entanglement entropy is spoiled by the appearance of quasirevivals which fade out in the thermodynamic limit, and which we interpret using a quasiparticle picture adapted to periodic drivings. We show that the time-averaged entanglement entropy in the synchronized regime obeys a volume law scaling with the subsystem's size. Such result is reminiscent of a thermal state or a generalized Gibbs ensemble, although the system does not heat up towards infinite temperature as a consequence of the integrability of the model.
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, scales with chain length
L as ${S}_{L/2}\ensuremath{\sim}lnL$; in contrast, for $\ensuremath{\gamma}>{\ensuremath{\gamma}}_{c}$, it becomes independent of
L. In the small $\ensuremath{\gamma}$ limit, we compute the coefficient, $\ensuremath{\alpha}$, of the
lnL 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 remain independent of
L 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 a class of periodically driven
d-dimensional integrable models and show that after
n 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
l 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
d (volume law) for large enough
l 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 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=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 ${S}_{\ensuremath{\infty}}$ as a function of $\ensuremath{\omega}$ whenever there is a crossing of the Floquet bands. We discuss experiments which can test our theory.