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arXiv:2111.13558 · Physical review. B./Physical review. B

Localization persisting under aperiodic driving

Hongzheng Zhao, Florian Mintert, Johannes Knolle, Roderich Moessner

Published Jun 6, 2022• 16 citations
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Abstract

Localization may survive in periodically driven (Floquet) quantum systems, but is generally unstable for aperiodic drives. In this Letter, we identify a hidden conservation law originating from a chiral symmetry in a disordered spin-$\frac{1}{2}$ XX chain. This protects indefinitely long-lived localization for general---even aperiodic---drives. Therefore, rather counterintuitively, adding further potential disorder which spoils the conservation law delocalizes the system, via a controllable parametrically long-lived prethermal regime. This provides an example of persistent single-particle ``localization without eigenstates.''

DOI
10.1103/physrevb.105.l220202

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Periodically driven systems often exhibit behavior distinct from static systems. In single-particle, static systems, any amount of disorder generically localizes all eigenstates in one dimension. In contrast, we show that in topologically nontrivial, single-particle Floquet loop drives with chiral symmetry in one dimension, a localization-delocalization transition occurs as the time ttt is varied within the driving period ($0\ensuremath{\le}t\ensuremath{\le}{T}_{\text{drive}}$). We find that the time-dependent localization length Lloc(t){L}_{\mathrm{loc}}(t)Lloc​(t) diverges with a universal exponent as ttt approaches the midpoint of the drive: ${L}_{\mathrm{loc}}(t)\ensuremath{\sim}{(t\ensuremath{-}{T}_{\text{drive}}/2)}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}=2$. We provide analytical and numerical evidence for the universality of this exponent within the AIII symmetry class.
Periodically driven classical many-body systems can host a rich zoo of prethermal dynamical phases. In this work, we extend the paradigm of classical prethermalization to aperiodically driven systems. We establish the existence of a long-lived prethermal regime in spin systems subjected to random multipolar drives. We demonstrate that the thermalization time scales as (1/T)^{2n+2}, where n is the multipolar order and T is the intrinsic time-scale associated with the drive. In the n→∞ limit, the drive becomes quasiperiodic and the thermalization time becomes exponentially long [∼exp(β/T)]. We further establish the robustness of prethermalization by demonstrating that these thermalization time scaling laws hold for a wide range of initial state energy densities. Intriguingly, the thermalization process in these classical systems is parametrically slower than their quantum counterparts, thereby highlighting important differences between classical and quantum prethermalization. Finally, we propose a protocol to harness this classical prethermalization to realize time rondeau crystals.
The ergodicity postulate, a foundational pillar of Gibbsian statistical mechanics predicts that a periodically driven (Floquet) system in the absence of any conservation law heats to a featureless `infinite temperature' state. Here, we find--for a clean and interacting generic spin chain subject to a {\it strong} driving field--that this can be prevented by the emergence of {\it approximate but stable} conservation-laws not present in the undriven system. We identify their origin: they do not necessarily owe their stability to familiar protections by symmetry, topology, disorder, or even high energy costs. We show numerically, {\it in the thermodynamic limit,} that when required by these emergent conservation-laws, the entanglement-entropy density of an infinite subsystem remains zero over our entire simulation time of several decades in natural units. We further provide a recipe for designing such conservation laws with high accuracy. Finally, we present an ensemble description, which we call the strongly driven ensemble incorporating these constraints. This provides a way to control many-body chaos through stable Floquet-engineering. Strong signatures of these conservation-laws should be experimentally accessible since they manifest in all length and time scales. Variants of the spin model we have used, have already been realized using Rydberg-dressed atoms.
It is by now well established that periodically driven quantum many-body systems can realize topological nonequilibrium phases without any equilibrium counterpart. Here we show that, even in the absence of time translation symmetry, nonequilibrium topological phases of matter can exist in aperiodically driven systems for tunably parametrically long prethermal lifetimes. As a prerequisite, we first demonstrate the existence of long-lived prethermal Anderson localization in two dimensions under random multipolar driving. We then show that the localization may be topologically nontrivial with a quantized bulk orbital magnetization even though there are no well-defined Floquet operators. We further confirm the existence of this anomalous random multipolar driven insulator by detecting quantized charge pumping at the boundaries, which renders it experimentally observable.
We investigate the electric-field driven power-law random banded matrix (PLRBM) model where a variation in the power-law exponent $\ensuremath{\alpha}$ yields a delocalization-to-localization phase transition. We examine the periodically driven PLRBM model with the help of the Floquet operator. The level spacing ratio and the generalized participation ratio of the Floquet Hamiltonian reveal a drive-induced weak multifractal (fractal) phase accompanied by diffusive (subdiffusive) transport on the delocalized side of the undriven PLRBM model. On the localized side, the time-periodic model remains localized---the average level-spacing ratio corresponds to Poisson statistics and logarithmic transport is observed in the dynamics. Extending our analysis to the aperiodic Thue-Morse driven system, we find that the aperiodically driven clean long-range hopping model (clean counterpart of the PLRBM model) exhibits the phenomenon of exact dynamical localization on tuning the drive parameters at special points. The disordered time-aperiodic system shows diffusive transport followed by relaxation to the infinite-temperature state on the delocalized side, and a prethermal plateau with subdiffusion on the localized side. Additionally, we compare this with a quasiperiodically driven Aubry-Andr\'e-Harper model that also undergoes a localization-delocalization transition. Unlike the disordered long-range model, it features a prolonged prethermal plateau followed by subdiffusion to the infinite temperature state, even on the delocalized side.
While a clean, driven system generically absorbs energy until it reaches "infinite temperature," it may do so very slowly exhibiting what is known as a prethermal regime. Here, we show that the emergence of an additional approximately conserved quantity in a periodically driven (Floquet) system can give rise to an analogous long-lived regime. This can allow for nontrivial dynamics, even from initial states that are at a high or infinite temperature with respect to an effective Hamiltonian governing the prethermal dynamics. We present concrete settings with such a prethernial regime, one with a period-doubled (time-crystalline) response. We also present a direct diagnostic to distinguish this prethermal phenomenon from its infinitely long-lived many-body localized cousin. We apply these insights to a model of the recent NMR experiments by Rovny et al. [Phys. Rev. Lett. 120, 180603 (2018)] which, intriguingly, detected signatures of a Floquet time crystal in a clean three-dimensional material. We show that a mild but subtle variation of their driving protocol can increase the lifetime of the time-crystalline signal by orders of magnitude.