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arXiv:2401.05697 · Physical review. B./Physical review. B

Dynamical chiral symmetry and symmetry-class conversion in Floquet topological insulators

Mohamed Assili, Panagiotis Kotetes

Published May 13, 2024• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.109.184307

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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 \ensuremathπ\ensuremath{\pi}\ensuremathπ 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 CI\ensuremath→AIII\mathrm{CI}\ensuremath{\rightarrow}\mathrm{AIII}CI\ensuremath→AIII transition, which now enables the system to harbor topological \ensuremathπ\ensuremath{\pi}\ensuremathπ 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 \ensuremathπ\ensuremath{\pi}\ensuremathπ modes.
Symmetry-protected topological (SPT) phases in insulators and superconductors are known for their robust edge modes, linked to bulk invariants through the bulk-boundary correspondence. While this principle traditionally applies to gapped phases, recent advances have extended it to gapless systems, where topological edge-states persist even in the absence of a bulk gap. We extend this framework to periodically driven chains with chiral symmetry, revealing the existence of topological edge zero and $\ensuremath{\pi}$ modes despite the lack of bulk gaps in the quasienergy spectrum. By examining the half-period decomposition of chiral evolutions, we construct topological invariants that circumvent the need to define the Floquet Hamiltonian, making them more suitable for generalization to the gapless regime. We provide explicit examples, including generalizations of the Kitaev chain and related spin models, where localized $\ensuremath{\pi}$ modes emerge even when the bulk is gapless at the same quasienergy as the edge modes. We numerically study the effect of interactions, which give a finite lifetime to the edge modes in the thermodynamic limit with the decay rate consistent with Fermi's golden rule.
M\"obius topological insulators have dispersive edge bands with M\"obius twists in momentum space, which are protected by the combination of chiral and Z2{\mathbb{Z}}_{2}Z2​-projective translational symmetries. In this work, we reveal a unique type of M\"obius topological insulator, whose edge bands can twist around the quasienergy $\ensuremath{\pi}$ of a periodically driven system and are thus of Floquet origin. By applying time-periodic quenches to an experimentally realized M\"obius insulator model, we obtain interconnected M\"obius edge bands around zero and $\ensuremath{\pi}$ quasienergies, which can coexist with a gapped or gapless bulk. These M\"obius bands are topologically characterized by a pair of generalized winding numbers, which are integer-quantized as a result of an emergent chiral symmetry at a high-symmetry point in momentum space. Numerical investigations of the quasienergy and entanglement spectra provide consistent evidence for the presence of such M\"obius topological phases. A protocol based on the adiabatic switching of edge-band populations is further introduced to dynamically characterize the topology of Floquet M\"obius edge bands. Our findings thus extend the scope of M\"obius topological phases to nonequilibrium settings and unveil a unique class of M\"obius-twisted topological edge states without static counterparts.
The existence of gapless boundary states is a key attribute of any topological insulator. Topological band theory predicts that these states are robust against static perturbations that preserve the relevant symmetries. In this article, using Floquet theory, we examine how chiral symmetry protection extends also to states subject to time-periodic perturbations---in one-dimensional Floquet topological insulators as well as in ordinary one-dimensional time-independent topological insulators. It is found that, in the case of the latter, the edge modes are resistant to a much larger class of time-periodic symmetry-preserving perturbations than in Floquet topological insulators. Notably, boundary states in chiral time-independent topological insulators also exhibit an unexpected resilience against a certain type of symmetry-breaking time-periodic perturbations. We argue that this is a generic property for topological phases protected by chiral symmetry. Implications for experiments are discussed.
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.
We explore the physics of a Chern insulator subjected to a two-step Floquet drive. We analytically obtain the phase diagram and show that the system can exhibit different topological phases characterized by the presence and chirality of edge modes in the two bulk gaps of the Floquet quasienergy spectrum, around 0 and $\ensuremath{\pi}$. We find that the phase of the system depends on the mean but not on the amplitude of the drive. The bulk topological invariants characterizing the phases can be extracted by mapping the unitary evolution within a time period to an energetically trivial but topologically nontrivial time evolution. An extensive numerical study of the bulk topological invariants in the presence of quenched disorder reveals transitions induced by strong disorder (i) from the different topological to trivial insulator phases and (ii) from a trivial to a topological Anderson insulator phase at intermediate disorder strengths. Careful analysis of level statistics of the quasienergy spectrum indicates a ``levitation-annihilation'' mechanism near these transitions.
Recently, anomalous Floquet topological phases without static counterparts have been observed in different systems, where periodically driven models are realized to support a winding number of 1 and a pair of edge modes in each quasienergy gap. Here, we focus on cold atomic gases in optical lattices and propose a driving scheme that breaks rotation symmetry but maintains inversion symmetry of the instantaneous Hamiltonian, and discover a type of anomalous Floquet topological phase with a winding number larger than 1. By analyzing the condition of band touching under the constraint of symmetries, we map out the phase diagram exactly by varying the driving parameters and discuss the quasienergy spectra of typical topological phases, which can present multiple pairs of edge modes within a single gap. Finally, we suggest to characterize the topology of such phases by detecting the band inversion surfaces via quench dynamics.