Entanglement entropy in a periodically driven quantum Ising ring

Abstract

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. DOI: 10.1103/PhysRevB.94.134304 The past decade has witnessed a remarkable revival of interest for the nonequilibrium dynamics of quantum manybody systems, mostly triggered by the advances in the experimental control of the tunable coupling, as well as of the environment isolation, of ultracold atomic lattices [1]. The possibility to observe the coherent time-resolved quantum dynamics of many-body systems has stimulated considerable theoretical interest in the different dynamical regimes of many-particle quantum systems, pioneered by studies on quantum quenches [2]. Albeit thermalization is eventually expected in ergodic systems [3], intermediate-time dynamics can display novel features determined by the nature of the driven out of equilibrium dynamics. A paradigmatic example of this scenario is given by periodically driven manybody systems [4], which can be traditionally employed to engineer hopping in optical lattices [5], design artificial gauge fields in cold atoms [6], or more recently to stabilize novel topological states of matter [7]. Such a plethora of promising applications has stimulated a substantial theoretical debate on the properties of many-body dynamics under periodic driving [8]. In the absence of a cooling mechanism, ergodic nonintegrable systems are expected to heat up towards an infinite temperature state [9], as a consequence of resonant energy absorption from the driving. However, even for an isolated periodically driven system, heating can occur on an extremely long time scale, and a richer scenario is expected at the intermediate stages of the dynamics, such as the emergence of novel prethermalized Floquet metastable states [10].At the same time, while entanglement entropy [11] is of key importance in condensed matter to characterize topological phases or to probe nonequilibrium dynamics [12], few works have addressed so far the evolution and the scaling of entanglement entropy in periodically driven quantum many-body systems [13]. In contrast, for a q...

Entanglement dynamics following a sudden quench: An exact solution

“…Using these expressions of b 1 , b 2 and Eqns. (20), (21) and (22), we can obtain the various Rényi entropies including the von Neumann entropy.…”

“…Following some initial works [13,14], a detailed analysis of the time development of the entanglement entropy in a quantum Ising chain under a quench in the transverse magnetic field was presented in [15]. Subsequently the phenomenon of entanglement dynamics has been investigated in the context of many-body localizations [16][17][18][19], spin chains [20] and rings [21], diffusive [22], integrable [15,23], nonintegrable [24] and various other discrete systems [25,26].…”

Fate of current, residual energy, and entanglement entropy in aperiodic driving of one-dimensional Jordan-Wigner integrable models

“…the volume of the subsystem 39,40 . The same kind of behavior in the EE is also observed for periodically driven one-dimensional quantum many-body systems, but the asymptotic value that the EE attains is in this case ascertained from a periodic steady state described by a periodic generalized Gibbs ensemble [41][42][43] . On the contrary, a disordered system or a state of a many body localized system has a characteristic logarithmic growth of entanglement in time [46][47][48][49][50][51] .…”

Two-rate periodic protocol with dynamics driven through many cycles

“…A block diagonal 2l d × 2l d correlation matrix C is constructed with l×l diagonal blocks 1−C and C and off-diagonal blocks F and F ⋆ respectively and then diagonalized to obtain the eigenvalues p i 's 29,30 . From there the entropy is obtained as…”