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arXiv:1603.06944 · Physical review. B./Physical review. B

Periodic table for Floquet topological insulators

Rahul Roy, Fenner Harper

Published Oct 13, 2017• 354 citations
arXivHTMLPDF

Abstract

Periodically driven systems have recently been shown to host topological phases that are inherently dynamical in character, opening up a new arena in which to explore topological physics. One important group of such phases, known as Floquet topological insulators, arise in systems of free fermions and exhibit protected topological edge modes analogous to the edge modes of static topological insulators. In this work, the authors use methods from K theory to provide a complete topological classification of Floquet topological phases of this kind. The main result is a periodic table for Floquet topological insulators, which may be viewed as a time-dependent extension of the periodic table of topological insulators and superconductors originally introduced by Alexei Kitaev.

DOI
10.1103/physrevb.96.155118

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Periodically driven systems have recently been shown to host topological phases that are inherently dynamical in character, opening up a new arena in which to explore topological physics. One important group of such phases, known as Floquet topological insulators, arise in systems of free fermions and exhibit protected topological edge modes analogous to the edge modes of static topological insulators. In this work, the authors use methods from K theory to provide a complete topological classification of Floquet topological phases of this kind. The main result is a periodic table for Floquet topological insulators, which may be viewed as a time-dependent extension of the periodic table of topological insulators and superconductors originally introduced by Alexei Kitaev.
This dissertation focus on the classification of topological matters in static and periodically driven systems.First, we build a complete topological classification of local unitary operators in free fermionic systems. This result can be encoded in a periodic table with a period of eight, similar to the periodic table of topological insulators. In our classification, we define two local unitaries in a certain symmetry class to be topologically equivalent if they can be connected via finite time evolutions (locally generated unitaries) without breaking the symmetries of the symmetry class while maintaining locality. In this way, the classification we derive will allow us to distinguish locally generated unitaries from those that cannot be locally generated.Besides, we also how to find possible generating Hamiltonians for locally generated unitaries. These results can be used to study the edge behaviors of Floquet systems.Second, we propose bulk invariants and edge invariants that are locally computable and improve existing topological invariants by being applicable to systems with the disorder. Afterward, we set up a rigorous connection between bulk and edge invariants for Floquet systems belonging to Class AIII and class AII of the Altland-Zirnbauer symmetry classification.Last, after treating a static system as a Floquet system generated by a constant Hamiltonian, we employ the tools we developed in Floquet systems to classify the edge behaviors in static systems.
Similar to static systems, periodically driven systems can host a variety of topologically non-trivial phases. Unlike the case of static Hamiltonians, the topological indices of bulk Floquet bands may fail to describe the presence and robustness of edge states, prompting the search for new invariants. We develop a unified description of topological phases and their invariants in driven systems, by using scattering theory. We show that scattering matrix invariants correctly describe the topological phase, even when all bulk Floquet bands are trivial. Additionally, we use scattering theory to introduce and analyze new periodically driven phases, such as weak topological Floquet insulators, for which invariants were previously unknown. We highlight some of their similarities with static systems, including robustness to disorder, as well as some of the features unique to driven systems, showing that the weak phase may be destroyed by breaking translational symmetry not in space, but in time.
Periodically driven (Floquet) phases are attractive due to their ability to host unique physical phenomena with no static counterparts. We propose a general approach in nontrivially devising a square-root version of existing Floquet phases, applicable both in noninteracting and in interacting setting. The resulting systems are found to yield richer physics that is otherwise absent in the original counterparts and is robust against parameter imperfection. These include the emergence of Floquet topological superconductors with arbitrarily many zero, $\ensuremath{\pi}$, and $\ensuremath{\pi}/2$ edge modes, as well as 4T4T4T-period Floquet time crystals in disordered and disorder-free systems (TTT being the driving period). Remarkably, our approach can be repeated indefinitely to obtain a 2nth{2}^{n}\mathrm{th}2nth-root version of any periodically driven system, thus, allowing for the discovery and systematic construction of exotic Floquet phases.
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.
Floquet insulators are periodically driven quantum systems that can host novel topological phases as a function of the drive parameters. These new phases exhibit features reminiscent of fermion doubling in discrete-time lattice fermion theories. We make this suggestion concrete by mapping the spectrum of a noninteracting (1+1)D Floquet insulator for certain drive parameters onto that of a discrete-time lattice fermion theory with a time-independent Hamiltonian. The resulting Hamiltonian is distinct from the Floquet Hamiltonian that generates stroboscopic dynamics. It can take the form of a discrete-time Su-Schrieffer-Heeger model with half the number of spatial sites of the original model, or of a (1+1)D Wilson-Dirac theory with one quarter of the spatial sites.
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.