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arXiv:2103.12107 · Physical review. B./Physical review. B

Dynamical fragile topology in Floquet crystals

Jiabin Yu, Yang Ge, S. Das Sarma

Published Nov 9, 2021• 9 citations
arXivHTMLPDF

Abstract

Although fragile topology has been intensely studied in static crystals in terms of Wannier obstruction, it is not clear how to generalize the concept to dynamical systems. In this work we generalize the concept of fragile topology, and provide a definition of fragile topology for noninteracting Floquet crystals, which we refer to as dynamical fragile topology. In contrast to the static fragile topology defined by Wannier obstruction, dynamical fragile topology is defined for the nontrivial quantum dynamics characterized by the obstruction to static limits (OTSL). Specifically, the OTSL of a Floquet crystal is fragile if and only if it disappears after adding a symmetry-preserving static Hamiltonian in a direct-sum way preserving the relevant gaps (RGs). We further present a concrete $2+1\mathrm{D}$ example for dynamical fragile topology, based on a model that is qualitatively the same as the dynamical model with anomalous chiral edge modes in Rudner et al. [Phys. Rev. X 3, 031005 (2013)]. The fragile OTSL in the $2+1\mathrm{D}$ example exhibits anomalous chiral edge modes for a natural open boundary condition, and does not require any crystalline symmetries besides lattice translations. Our work paves the way to study fragile topology for general quantum dynamics.

DOI
10.1103/physrevb.104.l180303

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Although fragile topology has been intensely studied in static crystals in terms of Wannier obstruction, it is not clear how to generalize the concept to dynamical systems. In this work we generalize the concept of fragile topology, and provide a definition of fragile topology for noninteracting Floquet crystals, which we refer to as dynamical fragile topology. In contrast to the static fragile topology defined by Wannier obstruction, dynamical fragile topology is defined for the nontrivial quantum dynamics characterized by the obstruction to static limits (OTSL). Specifically, the OTSL of a Floquet crystal is fragile if and only if it disappears after adding a symmetry-preserving static Hamiltonian in a direct-sum way preserving the relevant gaps (RGs). We further present a concrete 2+1D2+1\mathrm{D}2+1D example for dynamical fragile topology, based on a model that is qualitatively the same as the dynamical model with anomalous chiral edge modes in Rudner et al. [Phys. Rev. X 3, 031005 (2013)]. The fragile OTSL in the 2+1D2+1\mathrm{D}2+1D example exhibits anomalous chiral edge modes for a natural open boundary condition, and does not require any crystalline symmetries besides lattice translations. Our work paves the way to study fragile topology for general quantum dynamics.
Various exotic topological phases of Floquet systems have been shown to arise from crystalline symmetries. Yet, a general theory for Floquet topology that is applicable to all crystalline symmetry groups is still in need. In this work, we propose such a theory for (effectively) non-interacting Floquet crystals. We first introduce quotient winding data to classify the dynamics of the Floquet crystals with equivalent symmetry data, and then construct dynamical symmetry indicators (DSIs) to sufficiently indicate the inherently dynamical Floquet crystals. The DSI and quotient winding data, as well as the symmetry data, are all computationally efficient since they only involve a small number of Bloch momenta. We demonstrate the high efficiency by computing all elementary DSI sets for all spinless and spinful plane groups using the mathematical theory of monoid, and find a large number of different nontrivial classifications, which contain both first-order and higher-order 2+1D anomalous Floquet topological phases. Using the framework, we further find a new 3+1D anomalous Floquet second-order topological insulator (AFSOTI) phase with anomalous chiral hinge modes.
A universal feature of topological insulators is that they cannot be adiabatically connected to an atomic limit, where individual lattice sites are completely decoupled. This property is intimately related to a topological obstruction to constructing a localized Wannier function from Bloch states of an insulator. Here, we generalize this characterization of topological phases toward periodically driven systems. We show that nontrivial connectivity of hybrid Wannier centers in momentum space and time can characterize various types of topology in periodically driven systems, which include Floquet topological insulators, anomalous Floquet topological insulators with micromotion-induced boundary states, and gapless Floquet states realized with topological Floquet operators. In particular, nontrivial time dependence of hybrid Wannier centers indicates impossibility of continuous deformation of a driven system into an undriven insulator, and a topological Floquet operator implies an obstruction to constructing a generalized Wannier function which is localized in real and frequency spaces. Our results pave a way to a unified understanding of topological states in periodically driven systems as a topological obstruction in Floquet states.
We construct an anomalous Floquet topological phase on a negatively curved hyperbolic lattice. The model is a tight-binding Hamiltonian with a periodically repeated four-color edge-hopping sequence and a sublattice-staggered onsite potential step. The topological regime is reached near the limit in which a single hopping step transfers amplitude completely across an active edge, while the trivial regime is reached near the point where two full hops occur along an active edge during a single hopping step, returning the amplitude to its starting site. In finite open patches, the topological regime is characterized by bulk quasienergy gaps at 000 and πππ that are populated by in-gap states, in contrast to a trivial regime where these gaps remain empty. Using compact periodic lattices, we map the bulk 000 and πππ quasienergy gaps and identify the gapped regions connected to the trivial and anomalous open-boundary spectra. We diagnose the in-gap states as chiral boundary modes by their real-space dynamics. Finally, we introduce a small-boundary spectral-flow diagnostic based on punctured periodic hyperbolic lattices, which avoids the ambiguity associated with the extensive outer boundary of finite hyperbolic patches. This puncture-based diagnostic should be useful for studying other topological hyperbolic systems.
The Floquet time crystal, which breaks discrete time-translation symmetry, is an intriguing phenomenon in nonequilibrium systems. It is crucial to understand the rigidity and robustness of discrete time crystal (DTC) phases in a many-body system, and finding a precisely solvable model can pave a way for understanding of the DTC phase. Here, we propose and study a solvable spin chain model by mapping it to a Floquet superconductor through the Jordan-Wigner transformation. The phase diagrams of Floquet topological systems are characterized by topological invariants and tell the existence of anomalous edge states. The subharmonic oscillation, which is the typical signal of the DTC, can be generated from such edge states and protected by topology and solvability. We also examine the robustness of the DTC by adding symmetry-preserving and symmetry-breaking perturbations. Our results on the topologically protected DTC can provide a deep understanding of the DTC when generalized to other interacting or dissipative systems.
We extend the notion of fragile topology to periodically driven systems. We demonstrate driving-induced fragile topology in two different models, namely, the Floquet honeycomb model and the Floquet $\ensuremath{\pi}$-flux square-lattice model. In both cases, we discover a rich phase diagram that includes Floquet fragile topological phases protected by crystalline rotation or mirror symmetries, Floquet Chern insulators, and trivial atomic phases, with distinct boundary features. Remarkably, the transitions between different phases can be feasibly achieved by simply tuning the driving amplitudes, which is a unique feature of driving-enabled topological phenomena. Moreover, corner-localized fractional charges are identified as a ``smoking-gun'' signal of fragile topology in our systems. Our work paves the way for studying and realizing fragile topology in Floquet systems.
Topological phases of matter are defined by their nontrivial patterns of ground-state quantum entanglement, which is irremovable so long as the excitation gap and the protecting symmetries, if any, are maintained. Recent studies on noninteracting electrons in crystals have unveiled a peculiar variety of topological phases, which harbor nontrivial entanglement that can be dissolved simply by the the addition of entanglement-free, but charged, degrees of freedom. Such topological phases have a weaker sense of robustness than their conventional counterparts, and are therefore dubbed ``fragile topological phases.'' In this work, we show that fragile topology is a general concept prevailing beyond systems of noninteracting electrons. Fragile topological phases can generally occur when a system has a U(1)\mathrm{U}(1)U(1) charge conservation symmetry, such that only particles with one sign of the charge are physically allowed (e.g., electrons but not positrons). We demonstrate that fragile topological phases exist in interacting systems of both fermions and of bosons.