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arXiv:0904.2172 · Physical Review B

Exotic freezing of response in a quantum many-body system

Arnab Das

Published Nov 3, 2010• 163 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.82.172402

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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 (\ensuremathω)(\ensuremath{\omega})(\ensuremathω) regime. In a periodically driven classical thermodynamic system, freezing at high \ensuremathω\ensuremath{\omega}\ensuremathω occurs when 1/\ensuremathω1/\ensuremath{\omega}1/\ensuremathω is much smaller than the characteristic relaxation time of the system and hence the freezing always increases there as \ensuremathω\ensuremath{\omega}\ensuremathω is increased. Here, in the contrary, we see surprising nonmonotonic freezing behavior of the response with \ensuremathω\ensuremath{\omega}\ensuremathω, 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 \ensuremathω\ensuremath{\omega}\ensuremathω) 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.
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.
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.
A class of integrable models, such as the one-dimensional transverse-field Ising model, responds nonmonotonically to a periodic drive with respect to the driving parameters and freezes almost absolutely for certain combinations of the latter. In this paper, we go beyond the two-band structure of the Ising-like models studied previously and ask whether such an unusual nonmonotonic response and near-absolute freezing occur in integrable systems with a higher number of bands. To this end, we consider a tight-binding model for bilayer graphene subjected to an interlayer potential difference. We find that when the potential is driven periodically, the system responds nonmonotonically to variations in the driving amplitude V0{V}_{0}V0​ and frequency $\ensuremath{\omega}$ and shows near-absolute freezing for certain values of ${V}_{0}/\ensuremath{\omega}$. However, the freezing occurs only in the presence of a constant bias in the driving, i.e., when $V={V}^{\ensuremath{'}}+{V}_{0}cos\ensuremath{\omega}t$. When ${V}^{\ensuremath{'}}=0$, the freezing is switched off for all values of ${V}_{0}/\ensuremath{\omega}$. We support our numerical results with analytical calculations based on a rotating wave approximation. We also give a proposal to realize the driven bilayer system via ultracold atoms in an optical lattice, where the driving can be implemented by shaking the lattice.
Dynamical many-body freezing occurs in periodic transverse field-driven integrable quantum spin systems. Under freezing conditions, quantum dynamics causes practically infinite hysteresis in the drive response, maintaining its starting value. We find similar resonant freezing in the Lipkin-Meshkov-Glick (LMG) model. In the LMG, the freezing conditions in the driving field suppresses the heating postulated by the eigenstate thermalization hypothesis (ETH) by inducing dynamical many-body localization, or DMBL. This is in contrast to many-body localization (MBL), which requires disorder to suppress ETH. DMBL has been validated by the inverse participation ratio (IPR) of the quasistationary Floquet modes. Similarly to the TFIM, the LMG exhibits high-frequency localization only at freezing points. IPR localization in the LMG deteriorates with an inverse system size law at lower frequencies, which indicates heating to infinite temperature. Furthermore, adiabatically increasing frequency and amplitude from low values raises the Floquet state IPR in the LMG from nearly zero to unity, indicating a phase crossover. This occurrence enables a future technique to construct an MBL engine in clean systems that can be cycled by adjusting drive parameters only.
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.
For a class of integrable quantum many-body systems, symmetric AC driving can generically produce a steady DC response. We show how such dynamical freezing can be switched off, not by forcing the system to follow the (arbitrarily fast) driving field, but rather through a much slower but complete oscillation of each individual mode of the system at a frequency of its own, with the slowest mode exhibiting a divergent period. This switching can be controlled in detail, its sharpness depending on a particular parameter of the Hamiltonian. The phenomenon has a robust manifestation even in the few-body limit, perhaps the most promising setting for realisation within existing frameworks.