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arXiv:1112.6171 · Physical Review B

Transverse Ising chain under periodic instantaneous quenches: Dynamical many-body freezing and emergence of slow solitary oscillations

Sirshendu Bhattacharyya, Arnab Das, Subinay Dasgupta

Published Aug 8, 2012• 77 citations
arXivHTMLPDF

Abstract

We study the real-time dynamics of a quantum Ising chain driven periodically by instantaneous quenches of the transverse field between $+{\ensuremath{\Gamma}}_{0}$ and ${\ensuremath{\Gamma}}_{0}$ back and forth in equal intervals of time. Two interesting phenomena are reported and analyzed. (i) We observe dynamical many-body freezing (DMF), i.e., strongly nonmonotonic freezing of the response with respect to the driving parameters (pulse width and height) resulting from coherent suppression of dynamics of all quasiparticle modes. For certain combinations of the pulse height and the period, maximal freezing (DMF peaks) is observed, where a massive collapse of the entire Floquet spectrum occurs and the many-body system remains frozen extremely close to the initial state for all time. (ii) Second, away from the freezing peak, we observe the emergence of a distinct oscillation with a single nontrivial frequency, which can be much lower than the driving frequency. This remarkable slow oscillation involving many high-energy modes dominates the response in the limit of long observation time. We identify this slow oscillation as the unique survivor of destructive quantum interference between the many-body modes. The oscillation tends to decay algebraically with time to a constant value. All the key features are demonstrated analytically with numerical evaluations for specific results.

DOI
10.1103/physrevb.86.054410

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We study the real-time dynamics of a quantum Ising chain driven periodically by instantaneous quenches of the transverse field between +\ensuremathΓ0+{\ensuremath{\Gamma}}_{0}+\ensuremathΓ0​ and \ensuremathΓ0{\ensuremath{\Gamma}}_{0}\ensuremathΓ0​ back and forth in equal intervals of time. Two interesting phenomena are reported and analyzed. (i) We observe dynamical many-body freezing (DMF), i.e., strongly nonmonotonic freezing of the response with respect to the driving parameters (pulse width and height) resulting from coherent suppression of dynamics of all quasiparticle modes. For certain combinations of the pulse height and the period, maximal freezing (DMF peaks) is observed, where a massive collapse of the entire Floquet spectrum occurs and the many-body system remains frozen extremely close to the initial state for all time. (ii) Second, away from the freezing peak, we observe the emergence of a distinct oscillation with a single nontrivial frequency, which can be much lower than the driving frequency. This remarkable slow oscillation involving many high-energy modes dominates the response in the limit of long observation time. We identify this slow oscillation as the unique survivor of destructive quantum interference between the many-body modes. The oscillation tends to decay algebraically with time to a constant value. All the key features are demonstrated analytically with numerical evaluations for specific results.
We experimentally demonstrate the phenomenon of dynamical many-body freezing in a periodically driven Ising chain. Theoretically [A. Das, Phys. Rev. B 82, 172402 (2010)], for certain values of the drive parameters all fundamental degrees of freedom contributing to the response dynamics freeze for all time and for arbitrary initial state. Also, since the condition of freezing involves only the drive parameters and not the quantization of the momentum (i.e., the system size), our simulation with a small (3-spin) chain captures all salient features of the freezing phenomenon predicted for the infinite chain. Using optimal control techniques, we realize high-fidelity cosine-modulated drive, and observe nonmonotonic freezing of magnetization at specific frequencies of modulation. Time evolution of the excitations in momentum space has been tracked directly through magnetization measurements.
We show that when a quantum many-body system is subjected to coherent periodic driving, the response may exhibit exotic freezing behavior in high driving frequency $(\ensuremath{\omega})$ regime. In a periodically driven classical thermodynamic system, freezing at high $\ensuremath{\omega}$ occurs when $1/\ensuremath{\omega}$ is much smaller than the characteristic relaxation time of the system and hence the freezing always increases there as $\ensuremath{\omega}$ is increased. Here, in the contrary, we see surprising nonmonotonic freezing behavior of the response with $\ensuremath{\omega}$, showing curious peak-valley structure. Quite interestingly, the entire system tends to freeze almost absolutely (the freezing peaks) when driven with a certain combination of driving parameters values (amplitude and $\ensuremath{\omega}$) due to coherent suppression of dynamics of the quantum many-body modes, which has no classical analog. We demonstrate this new freezing phenomenon analytically (supported by large-scale numerics) for a general class of integrable quantum spin systems.
Dynamical localization is one of the most startling manifestations of quantum interference, where the evolution of a simple system is frozen out under a suitably tuned coherent periodic drive. Here we show that, although any randomness in the interactions of a many-body system kills dynamical localization eventually, spectacular remnants survive even when the disorder is strong. We consider a disordered quantum Ising chain where the transverse magnetization relaxes exponentially with time with a decay time-scale $\ensuremath{\tau}$ due to random longitudinal interactions between the spins. We show that, under external periodic drive, this relaxation slows down ($\ensuremath{\tau}$ shoots up) by orders of magnitude as the ratio of the drive frequency $\ensuremath{\omega}$ and amplitude h0{h}_{0}h0​ tends to certain specific values (the freezing condition). If $\ensuremath{\omega}$ is increased while maintaining the ratio ${h}_{0}/\ensuremath{\omega}$ at a fixed freezing value, then $\ensuremath{\tau}$ diverges exponentially with $\ensuremath{\omega}.$ The results can be easily extended for a larger family of disordered fermionic and bosonic systems.
This work investigates dynamical quantum phase transitions (DQPTs) in a one-dimensional Ising model subjected to a periodically modulated transverse field. In contrast to sudden quenches, we demonstrate that a DQPT can be induced in two distinct ways. First, when the system remains within a given phase--ferromagnetic (FM) or paramagnetic (PM), a resonant periodic drive can trigger a DQPTs when its frequency matches the energy-level transition of the system. This DQPT is intimately connected to the emergence of Floquet topological phases. The timescale for the transition is governed by the perturbation strength λ′λ'λ′, the critical mode kck_ckc​, and its energy gap ΔkcΔ_{k_c}Δkc​​, following the scaling relation τ∝Δkcλ′−1csc⁡kcτ\proptoΔ_{k_c}λ'^{-1}\csc k_cτ∝Δkc​​λ′−1csckc​. Second, for drives across the critical point between the FM and PM phases, low frequencies can always induce DQPT, regardless of resonance. This behavior stems from the degeneracy of the energy-level at the critical point, which ensures that any drive with a frequency lower than the system's intrinsic transition frequency will inevitably excite the system. However, in the high-frequency regime, such excitation will be strongly suppressed, thereby inhibiting the occurrence of DQPTs. This study provides deeper insight into the nonequilibrium dynamics of quantum spin chains.
We study the coherent dynamics of a quantum many-body system subject to a time-periodic driving. We argue that in many cases, destructive interference in time makes most of the quantum averages time periodic, after an initial transient. We discuss in detail the case of a quantum Ising chain periodically driven across the critical point, finding that, as a result of quantum coherence, the system never reaches an infinite temperature state. Floquet resonance effects are moreover observed in the frequency dependence of the various observables, which display a sequence of well-defined peaks or dips. Extensions to nonintegrable systems are discussed.
Periodically driven Floquet quantum many-body systems have revealed new insights into the rich interplay of thermalization and growth of entanglement. The phenomenology of , whereby a translationally invariant many-body system exhibits emergent conservation laws and a slow growth of entanglement entropy at certain fixed ratios of a drive amplitude and frequency, presents a novel paradigm for retaining memory of an initial state up to late times. Previous studies of dynamical freezing have largely been restricted to a high-frequency Floquet-Magnus expansion and numerical exact diagonalization. Both techniques are unable to capture the slow approach to thermalization, or lack thereof, in a systematic fashion. By employing Floquet , where the time-dependent part of the Hamiltonian is gradually decoupled from the effective Hamiltonian using a sequence of unitary transformations, we unveil the universal approach to dynamical freezing and beyond, at asymptotically late times. We analyze the behavior associated with the flow renormalization at and near freezing using both exact-diagonalization and tensor-network-based methods and contrast the results with the conventional prethermal phenomenon. For a generic nonintegrable spin Hamiltonian with a periodic cosine wave drive, the flow approaches an unstable fixed point with an approximate emergent symmetry. We observe that at freezing the thermalization timescales are delayed compared to away from freezing, and the flow trajectory undergoes a series of events. Our numerical results are supported by analytical solutions to the flow equations.