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 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 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.
We study the dynamics of a periodically driven Rydberg chain starting from the state with zero Rydberg excitations (vacuum state denoted by $|0\ensuremath{\rangle}$) using a square pulse protocol in the high drive amplitude limit. We show, using exact diagonalization for finite system sizes $(L\ensuremath{\le}26)$, that the Floquet Hamiltonian of the system, within a range of drive frequencies that we chart out, hosts a set of quantum scars that have large overlap with the $|0\ensuremath{\rangle}$ state. These scars are distinct from their counterparts having high overlap with the maximal Rydberg excitation state $(|{\mathbb{Z}}_{2}\ensuremath{\rangle})$; they coexist with the latter class of scars and lead to persistent coherent oscillations of the density-density correlator starting from the $|0\ensuremath{\rangle}$ state. We also identify special drive frequencies at which the system undergoes perfect dynamic freezing, and we provide an analytic explanation for this phenomenon. Finally, we demonstrate that for a wide range of drive frequencies, the system reaches a steady state with subthermal values of the density-density correlator. The presence of such subthermal steady states, which are absent for dynamics starting from the $|{\mathbb{Z}}_{2}\ensuremath{\rangle}$ state, imply a weak violation of the eigenstate thermalization hypothesis in finite-sized Rydberg chains distinct from that due to the scar-induced persistent oscillations reported earlier. We conjecture that in the thermodynamic limit, such states may exist as prethermal steady states that show anomalously slow relaxation. We supplement our numerical results by deriving an analytic expression for the Floquet Hamiltonian using a Floquet perturbation theory in the high amplitude limit, which provides an analytic, albeit qualitative, understanding of these phenomena at arbitrary drive frequencies. We discuss experiments that can test our theory.
We study the stroboscopic dynamics of a periodically driven finite Rydberg chain with staggered ($\mathrm{\ensuremath{\Delta}}$) and time-dependent uniform $[\ensuremath{\lambda}(t)]$ detuning terms using exact diagonalization. We show that at intermediate drive frequencies (${\ensuremath{\omega}}_{D}$), the presence of a finite $\mathrm{\ensuremath{\Delta}}$ results in violation of the eigenstate thermalization hypothesis (ETH) via clustering of Floquet eigenstates. Such clustering is lost at special commensurate drive frequencies for which $\ensuremath{\hbar}{\ensuremath{\omega}}_{d}=n\mathrm{\ensuremath{\Delta}}$ ($n\ensuremath{\in}Z$) leading to restoration of ergodicity. The violation of ETH in these driven finite-sized chains is also evident from the dynamical freezing displayed by the density-density correlation between Rydberg excitations at even sites of the chain for specific ${\ensuremath{\omega}}_{D}$. Such a correlator exhibits stable oscillations with perfect revivals when driven close to the freezing frequencies for initial all spin-down ($|0\ensuremath{\rangle}$) or Neel ($|{\mathbb{Z}}_{2}\ensuremath{\rangle}$, with up spins on even sites) states. In contrast, for the $|{\overline{\mathbb{Z}}}_{2}\ensuremath{\rangle}$ (time-reversed partner of $|{\mathbb{Z}}_{2}\ensuremath{\rangle}$) initial state, we find complete absence of such oscillations leading to freezing for a range of ${\ensuremath{\omega}}_{D}$; this range increases with $\mathrm{\ensuremath{\Delta}}$. We also study the properties of quantum many-body scars in the Floquet spectrum of the model as a function of $\mathrm{\ensuremath{\Delta}}$ and show the existence of mid-spectrum scars at large $\mathrm{\ensuremath{\Delta}}$ which do not have overlap with either $|0\ensuremath{\rangle}$ or $|{\mathbb{Z}}_{2}\ensuremath{\rangle}$ states. We supplement our numerical results with those from an analytic Floquet Hamiltonian computed using Floquet perturbation theory which allows us to provide qualitative analytical explanations of the above-mentioned numerical results.
We study the periodic dynamics of a spin-one chain driven using a square-pulse protocol with amplitude
Q0 and frequency
ωD. The Hamiltonian of the spin chain hosts a thermodynamically large number of
Z2-valued conserved quantities
Wℓ on the links
ℓ. This allows us to study the Floquet dynamics of this chain within a given sector with fixed values of
Wℓ. For the sector with all
Wℓ=1, we find signatures of quantum many-body scar states for
ℏωD≫Q0; they lead to oscillatory dynamics and fidelity revival for specific initial states. Upon lowering
ωD, we find an ergodic regime exhibiting fast thermalization consistent with the prediction of the (Floquet) eigenstate thermalization hypothesis. In addition, we identify special drive frequencies
ωn∗=Q0/(2nℏ) (where
n=1,2,3,⋯) at which the Floquet Hamiltonian exhibits prethermal strong Hilbert space fragmentation (HSF) with the largest fragment being ergodic; in contrast, a weak HSF is found at
ωn′=Q0/[ℏ(2n+1)] (where
n=0,1,2,⋯). We also study the sector with
Wℓ={⋯1,1,−1,1,1,−1⋯} which shows strong HSF at
ωn∗ but no fragmentation at
ωn′. Our analysis indicates that the strong HSF in this sector harbors an integrable largest fragment. We provide numerical support for our analytical and perturbative results using exact-diagonalization (ED) studies on finite chains of length
L≤24. Our numerical results for entanglement entropy, fidelity, and correlation functions of the driven chain provide definitive signatures of prethermal strong HSF for both sectors.
We study the prethermal Floquet phases of a two-dimensional Rydberg atom array on a rectangular lattice in the presence of a periodic drive with large drive amplitude. We derive an analytic, albeit perturbative, Floquet Hamiltonian using Floquet perturbation theory (FPT) which charts out these phases and shows that the transition between them can be accessed by tuning the drive frequency. Using both numerical exact diagonalization on finite-size arrays and an analytical first-order Floquet Hamiltonian derived using FPT, we show that these prethermal Floquet phases and the transitions between them can be detected by studying the dynamics of equal-time density-density correlation functions of the Rydberg atoms. Our analysis thus provides a simple way of detecting these phases and associated transitions in this system; such a detection can be achieved in standard experiments, which we discuss.