Entanglement generation in periodically driven integrable systems: Dynamical phase transitions and steady state

Abstract

We study a class of periodically driven d−dimensional integrable models and show that after n drive cycles with frequency ω, pure states with non-area-law entanglement entropy Sn(l) ∼ l α (n,ω) are generated, where l is the linear dimension of the subsystem, and d − 1 ≤ α(n, ω) ≤ d. We identify and analyze the crossover phenomenon from an area (S ∼ l d−1 for d ≥ 1) to a volume (S ∼ 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 Sn generically decays to S∞ as (ω/n) (d+2)/2 for fast and (ω/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 ω 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∞ as a function of ω whenever there is a crossing of the Floquet bands. We discuss experiments which can test our theory.

Interconnections between equilibrium topology and dynamical quantum phase transitions in a linearly ramped Haldane model

“…Henceforth, experimental studies have been performed to probe intriguing dynamical phenomena like the real time evolution of closed quantum systems in cold atomic gases, [3], prethermalization [4][5][6], light-cone like propagation of quantum correlations [7], light-induced non-equilibrium superconductivity and topological systems [8,9] and many-body localization in disordered interacting systems [10]. In parallel, there have been a plethora of theoretical works on e.g., the growth of entanglement entropy following a quench [11], thermalization [12], light-induced topological matters [13][14][15], dynamics of topologically ordered systems [16][17][18], periodically driven closed quantum systems [19][20][21][22][23] and many body localization [24,25], to name a few. (For review, we refer to [26][27][28][29][30][31].…”

Coexistence of Different Scaling Laws for the Entanglement Entropy in a Periodically Driven System

“…We refer the reader to Ref. [12] for the dynamical behaviour of S(l; nT), whereas in the following section we show that different scaling laws for the EE coexist at finite times.…”

“…In between l 1 and l 2 a transitory region where neither a volume nor a area (log) law applies. Note that, notwihstanding the increase of the EE under periodic drive [12], the latter occurs in such a way to fulfill the log law and, more importantly, it obeys the finite size scaling Ansatz [19] typical of critical systems at equilibrium Fig. 2.…”

“…Recently, the out-of-equilibrium dynamics of many-body systems has received considerable attention [8]. As a consequence, also the dynamics of the EE of a quantum many-body system brought out-of-equilibrium by means of a quench [9][10][11] or a periodic drive has been throughout investigated [12][13][14]. A universal behaviour of the EE, i.e., independent of the details of the protocol pushing the system out of equilibrium, has been observed.…”

Equilibration and order in quantum Floquet matter

“…It is worth noting that the PGE density matrix leads to a volume law entanglement entropy that is less than the infinite temperature value, thus confirming a lack of heating. 34 The moral of this part of the story is that much less information survives in the free fermion late time states than does in the diagonal ensembles that describe Floquet-MBL systems but more than survives for the Floquet-ETH case. Recent developments and outlook: In a flurry of work, the program of identifying stable interacting Floquet phases has been pushed quite far already.…”