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arXiv:1704.00782 · Physical review. B./Physical review. B

Robustness of symmetry-protected topological states against time-periodic perturbations

Oleksandr Balabanov, Henrik Johannesson

Published Jul 27, 2017• 34 citations
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Abstract

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.

DOI
10.1103/physrevb.96.035149

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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.
It is by now a well known fact that boundary states in conventional time-independent\ntopological insulators are protected against perturbations that preserve relevant\nsymmetries. In the first part of this thesis, accompanying Papers A - C, we study how\nthis robustness extends to time-periodic (Floquet) topological insulators. Floquet\ntheory allows us to go beyond ordinary time-independent perturbations and study\nalso periodically-driven perturbations of the boundary states. The time-dependence\nhere opens up an extra lever of control and helps to establish the robustness to a\nmuch broader class of perturbations. In Paper A, a general idea behind the topological\nprotection of the boundary states against time-periodic perturbations is\npresented. In Paper B we address the experimental detection of the proposed robustness\nand suggest that signatures of it can be seen in the measurements of linear\nconductance. Our idea is explicitly illustrated on a case study: A topologically nontrivial\narray of dimers weakly attached to external leads. The discussed features are\ndescribed analytically and confirmed numerically. All computations are performed\nby employing a convenient methodology developed in Paper C. The idea is to combine\nLandauer-Büttiker theory with the so-called Floquet-Sambe formalism. It is\nshown that in this way all formulas for currents and densities essentially replicate\nwell known expressions from time-independent theory.\nTo find closed mathematical expressions for topological indices is in general a\nnontrivial task, especially in presence of various symmetries and/or interactions.\nThe second part of the thesis introduces a computational protocol, based on artificial\nneural networks and a novel topological augmentation procedure, capable of finding\ntopological indices with minimal external supervision. In Paper D the protocol is\npresented and explicitly exemplified on two simple classes of topological insulators\nin 1d and 2d. In Paper E we significantly advance the protocol to the classification\nof a more general type of systems. Our method applies powerful machine-learning\nalgorithms to topological classification, with a potential to be extended to more\ncomplicated classes where known analytical methods may become inapplicable.\nThe thesis is meant to serve as a supplement to the work contained in Papers A-E.\nHere we provide an extensive introduction to Floquet theory, focused on developing\nthe machinery for describing time-periodic topological insulators. The basic theory\nof artificial neural nets is also presented.
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.
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.
Symmetry plays an important role in the topological band theory. Recently, the concept of subsymmetry-protected topological (sub-SPT) phases [] suggests that chiral edge states can exist in a system with broken symmetry. However, sub-SPT theory only focuses on the zero-energy edge states in the half-filling gap, which limits its application. In this work, we extend the scope of sub-SPT phases in both one-dimensional (1D) and two-dimensional (2D) multigap asymmetric systems through redefined sublattices. In three typical asymmetric models without subsymmetry, we find that all the nonzero edge (corner) states can be illustrated by sub-SPT chiral edge (corner) states after redefining sublattices. Moreover, these nonzero chiral boundary states can exhibit some exotic properties, such as the topological bound states in the continuum. As to the bulk-edge correspondence, a universal topological invariant Z ¯ is defined, which can precisely predict the appearance of chiral boundary states in both 1D and 2D systems. Finally, we propose an acoustic experiment where the nonzero chiral corner states are numerically observed. Our work exhibits a general approach to study the topological properties of asymmetric systems, opening up possibilities for topological phenomena.
Topologically protected edge states exactly at topological phase boundaries challenge the conventional belief that topological states must be associated with a bulk energy gap. Because periodically driven (Floquet) systems host unusually intricate topological phase boundaries, topological edge states can be prolific at such Floquet quantum criticality. Working on a class of chiral-symmetric, Floquet-driven Majorana fermion chains, we analytically and computationally show that the precise boundaries between different Floquet topological gapped phases can accommodate topological edge modes, including the so-called critical Majorana π modes. We also identify a general bulk-edge correspondence formula to predict and understand the emergence of topological edge modes at Floquet quantum criticality. Of direct interest to quantum simulation experiments, our work opens an avenue for exploring topological physics intrinsic to nonequilibrium phase boundaries. The authors develop a theoretical framework to characterize the topology and edge states at the nonequilibrium critical points of periodically driven (Floquet) quantum systems. The rule of bulk-edge correspondence applicable to both gapped and gapless Floquet topological phases in 1D chiral symmetric driven systems is proposed.