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arXiv:2112.03167 · Physical review. B./Physical review. B

Extrinsic topology of Floquet anomalous boundary states in quantum walks

Takumi Bessho, Ken Mochizuki, Hideaki Obuse, Masatoshi Sato

Published Mar 14, 2022• 6 citations
arXivHTMLPDF

Abstract

Bulk-boundary correspondence is a fundamental principle for topological phases where bulk topology determines gapless boundary states. On the other hand, it has been known that corner or hinge modes in higher-order topological insulators may appear due to ``extrinsic'' topology of the boundaries even when the bulk topological numbers are trivial. In this paper, we find that Floquet anomalous boundary states in quantum walks have similar extrinsic topological natures. In contrast to higher-order topological insulators, the extrinsic topology in quantum walks is manifest even for first-order topological phases. We present the topological table for extrinsic topology in quantum walks and illustrate extrinsic natures of Floquet anomalous boundary states in concrete examples.

DOI
10.1103/physrevb.105.094306

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Bulk-boundary correspondence is a fundamental principle for topological phases where bulk topology determines gapless boundary states. On the other hand, it has been known that corner or hinge modes in higher-order topological insulators may appear due to ``extrinsic'' topology of the boundaries even when the bulk topological numbers are trivial. In this paper, we find that Floquet anomalous boundary states in quantum walks have similar extrinsic topological natures. In contrast to higher-order topological insulators, the extrinsic topology in quantum walks is manifest even for first-order topological phases. We present the topological table for extrinsic topology in quantum walks and illustrate extrinsic natures of Floquet anomalous boundary states in concrete examples.
Floquet non-Abelian topological phases emerge in periodically driven systems and exhibit properties that are absent in their Abelian or static counterparts. Dubbed the Floquet non-Abelian topological insulators (FNATIs), they are characterized by non-Abelian topological charges and feature multifold bulk-boundary correspondence, making their experimental observation challenging. Here we simulate the FNATI using a higher-dimensional photonic quantum walk and develop dynamic measurement schemes to demonstrate key signatures of the FNATI. Importantly, combining a direct bulk-dynamic detection for the underlying quaternion topological charge, and a spatially-resolved injection spectroscopy for the edge states, we experimentally establish the multifold bulk-boundary correspondence, and, in particular, identify the anomalous non-Abelian phase where edge states appear in all band gaps, despite the presence of a trivial topological charge. Our experiment marks the first experimental characterization of the FNATI, providing general insight into the non-Abelian topological phases.
The topological characterization of nonequilibrium topological matter is highly nontrivial because familiar approaches designed for equilibrium topological phases may not apply. In the presence of crystal symmetry, Floquet topological insulator states cannot be easily distinguished from normal insulators by a set of symmetry eigenvalues at high-symmetry points in the Brillouin zone. This work advocates a physically motivated, easy-to-implement approach to enhance the symmetry analysis to distinguish between a variety of Floquet topological phases. Using a two-dimensional inversion-symmetric periodically driven system as an example, we show that the symmetry eigenvalues for anomalous Floquet topological states, of both first order and second order, are the same as for normal atomic insulators. However, the topological states can be distinguished from one another and from normal insulators by inspecting the occurrence of stable dynamical symmetry inversion points in their microscopic dynamics. The analysis points to a position-space picture for understanding how topological boundary states can coexist with localized bulk states in anomalous Floquet topological phases.
Dynamical quantum phase transitions occur in dynamically evolving quantum systems when nonanalyticities occur at critical times in the return rate, a dynamical analog of the free energy. This extension of the concept of phase transitions can be brought into contact with another, namely, that of topological phase transitions in which the phase transition is marked by a change in a topological invariant. Following a quantum quench dynamical quantum phase transitions can happen in topological matter, a fact which has already been explored in one-dimensional topological insulators and in two-dimensional Chern insulators. Additionally, in one-dimensional systems a dynamical bulk boundary correspondence has been seen, related to the periodic appearance of zero modes of the Loschmidt echo itself. Here we extend both of these concepts to two-dimensional higher-order topological matter in which the topologically protected boundary modes are corner modes. We consider a minimal model which encompasses all possible forms of higher-order topology in two-dimensional topological band structures. We find that dynamical quantum phase transitions can still occur, and can occur for quenches which cross both bulk and boundary gap closings. Furthermore, a dynamical bulk boundary correspondence is also found, which takes a different form to that in one dimension.
Higher-order topological insulators not only exhibit the exotic bulk-boundary correspondence principle but also have an important application in quantum computing. However, it seems they have not been achieved in the discrete-time quantum walk. In this paper we construct a two-dimensional coinless discrete-time quantum walk to simulate the second-order topological insulator with zero-dimensional corner states. We show that both of the corner and edge states can be observed through the probability distribution of the walker after a multistep coinless discrete-time quantum walk. Finally, we propose a possible experimental implementation to realize this coinless discrete-time quantum walk in a three-dimensional integrated photonic circuit and verify the robustness of the corner states against the small disorder. Our work offers an alternative route to explore exotic higher-order topological matters using discrete-time quantum walks.
While periodically driven phases offer a unique insight into nonequilibrium topology that is richer than its static counterpart, their experimental realization is often hindered by ubiquitous decoherence effects. Recently, we have proposed a decoherence-free approach of realizing these Floquet phases. The central insight is that the reflection matrix, being unitary for a bulk insulator, plays the role of a Floquet time-evolution operator. We have shown that reflection processes off the boundaries of systems supporting higher-order topological phases (HOTPs) simulate nontrivial Floquet phases. So far, this method was shown to work for one-dimensional Floquet topological phases protected by local symmetries. Here, we extend the range of applicability by studying reflection off three-dimensional HOTPs with corner and hinge modes. We show that the reflection processes can simulate both first-order and second-order Floquet phases, protected by a combination of local and spatial symmetries. For every phase, we discuss appropriate topological invariants calculated with the nested scattering matrix method.