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arXiv:2310.20696 · Physical review. B./Physical review. B

Universal localization-delocalization transition in chiral-symmetric Floquet drives

Adrian B. Culver, Pratik Sathe, Albert Brown, Fenner Harper, Rahul Roy

Published Jul 16, 2025• 1 citation
arXivHTMLPDF

Abstract

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 $t$ is varied within the driving period ($0\ensuremath{\le}t\ensuremath{\le}{T}_{\text{drive}}$). We find that the time-dependent localization length ${L}_{\mathrm{loc}}(t)$ diverges with a universal exponent as $t$ 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.

DOI
10.1103/ntl6-9zlt

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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≤t\ensuremath≤Tdrive0\ensuremath{\le}t\ensuremath{\le}{T}_{\text{drive}}0\ensuremath≤t\ensuremath≤Tdrive​). 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: Lloc(t)\ensuremath∼(t\ensuremath−Tdrive/2)\ensuremath−\ensuremathν{L}_{\mathrm{loc}}(t)\ensuremath{\sim}{(t\ensuremath{-}{T}_{\text{drive}}/2)}^{\ensuremath{-}\ensuremath{\nu}}Lloc​(t)\ensuremath∼(t\ensuremath−Tdrive​/2)\ensuremath−\ensuremathν with \ensuremathν=2\ensuremath{\nu}=2\ensuremathν=2. We provide analytical and numerical evidence for the universality of this exponent within the AIII symmetry class.
We investigate the effects of disorder on a periodically driven one-dimensional model displaying quantized topological transport. We show that, while instantaneous eigenstates are necessarily Anderson localized, the periodic driving plays a fundamental role in delocalizing Floquet states over the whole system, henceforth allowing for a steady-state nearly quantized current. Remarkably, this is linked to a localization-delocalization transition in the Floquet states at strong disorder, which occurs for periodic driving corresponding to a nontrivial loop in the parameter space. As a consequence, the Floquet spectrum becomes continuous in the delocalized phase, in contrast with a pure-point instantaneous spectrum.
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-12\frac{1}{2}21​ 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.''
In this work, we discuss properties with no static counterpart arising in Floquet topological insulators with a dynamical chiral symmetry (DCS), i.e., a chiral symmetry which is present while driving. We explore the topological properties of Floquet insulators possessing a DCS which either does or does not survive upon taking the static limit. We consider the case of harmonic drives and employ a general framework using the quasienergy operator in frequency space. We find that for a DCS with no static analog, the presence of driving has a negligible impact on the topological phases associated with zero quasienergy. In stark contrast, topological gaps can open at $\ensuremath{\pi}$ quasienergy and mainly occur at momenta where the driving perturbation vanishes. We confirm the above general predictions for an extended Kitaev chain model in the BDI symmetry class. Another possibility that opens up when adding the drive, while preserving chiral symmetry, is symmetry-class conversion. We demonstrate such an effect for a static CI class Hamiltonian which is topologically trivial in one dimension. By considering a suitable driving, we obtain a $\mathrm{CI}\ensuremath{\rightarrow}\mathrm{AIII}$ transition, which now enables the system to harbor topological $\ensuremath{\pi}$ modes. Notably, the arising topological phases strongly depend on whether or not the DCS has a static analog. Our results bring Floquet insulators with nonstandard DCS forward as ideal candidate platforms for engineering and manipulating topological $\ensuremath{\pi}$ modes.
We study the localization-delocalization transition of Floquet eigenstates in a driven fermionic chain with an incommensurate Aubry-Andr\'e potential and a hopping amplitude which is varied periodically in time. Our analysis shows the presence of a mobility edge separating single-particle delocalized states from localized and multifractal states in the Floquet spectrum. Such a mobility edge does not have any counterpart in the static Aubry-Andr\'e model and exists for a range of drive frequencies near the critical frequency at which the transition occurs. The presence of the mobility edge is shown to leave a distinct imprint on fermion transport in the driven chain; it also influences the Shannon entropy and the survival probability of the fermions at long times. In addition, we find the presence of CAT (linear superposition of localized single particle states) states in the Floquet spectrum with weights centered around a few nearby sites of the chain. This is shown to be tied to the flattening of Floquet bands over a range of quasienergies. We support our numerical studies with a semianalytic expression for the Floquet Hamiltonian (HF{H}_{F}HF​) computed within a Floquet perturbation theory. The eigenspectra of the perturbative HF{H}_{F}HF​ so obtained exhibit qualitatively identical properties to the exact eigenstates of HF{H}_{F}HF​ obtained numerically. Our results thus constitute an analytic expression of a HF{H}_{F}HF​ whose spectrum supports multifractal and cat states. We suggest experiments which can test our theory.
Subjecting a many-body localized system to a time-periodic drive generically leads to delocalization and a transition to ergodic behavior if the drive is sufficiently strong or of sufficiently low frequency. Here we show that a specific drive can have an opposite effect, taking a static delocalized system into the many-body localized phase. We demonstrate this effect using a one-dimensional system of interacting hard-core bosons subject to an oscillating linear potential. The system is weakly disordered, and is ergodic absent the driving. The time-periodic linear potential leads to a suppression of the effective static hopping amplitude, increasing the relative strengths of disorder and interactions. Using numerical simulations, we find a transition into the many-body localized phase above a critical driving frequency and in a range of driving amplitudes. Our findings highlight the potential of driving schemes exploiting the coherent destruction of tunneling for engineering long-lived Floquet phases.
Periodic driving provides a powerful route for controlling localization and engineering effective Hamiltonians in quantum systems. A localized one-dimensional Aubry-André chain can be driven into a fully delocalized phase when the entire system is driven below a critical frequency. Here, we ask whether such delocalization can be achieved by driving only a part of the system. We address this question using a two-leg ladder comprising an Aubry-André chain coupled to a periodically driven clean tight-binding chain. Exact numerical calculations show that, starting from a localized static ladder, subsystem driving below a critical frequency cannot achieve complete delocalization. Instead, a mixed phase emerges, with localized and delocalized states coexisting and the fraction of delocalized states varying nonmonotonically with the driving frequency. This behavior can be understood using local effective Hamiltonians. Although these effective Hamiltonians are derived analytically in the high-frequency regime, they can nevertheless predict the localization transition occurring in the moderate-frequency regime. In fact, by quantifying the emergent nonlocality of the Floquet Hamiltonian, we show that these effective Hamiltonians provide a valid description up to a crossover frequency that is lower than the critical frequency associated with the localization transition. Our results demonstrate an alternative route to Floquet engineering, in which the transport properties of a quantum lattice system can be modified by driving only a subsystem.