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arXiv:1708.01173 · Mathematical Physics Analysis and Geometry

Topological Boundary Invariants for Floquet Systems and Quantum Walks

Christian Sadel, Hermann Schulz‐Baldes

Published Oct 6, 2017• 31 citations
arXivHTMLPDF
DOI
10.1007/s11040-017-9253-1

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Discrete-time quantum walks have been shown to simulate all known topological phases in one and two dimensions. Being periodically driven quantum systems, their topological description, however, is more complex than that of closed Hamiltonian systems. We map out the topological phases of the particle-hole symmetric one-dimensional discrete-time quantum walk. We find that there is no chiral symmetry in this system: its topology arises from the particle-hole symmetry alone. We calculate the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological invariant in a simple way that is consistent with a general definition for one-dimensional periodically driven quantum systems. These results allow for a transparent interpretation of the edge states on a finite lattice via the the bulk-boundary correspondence. We find that the bulk Floquet operator does not contain all the information needed for the topological invariant. As an illustration to this statement, we show that in the split-step quantum walk, the edges between two bulks with the same Floquet operator can host topologically protected edge states.
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.
One-dimensional discrete-time quantum walks show a rich spectrum of topological phases that have so far been exclusively analyzed using the Floquet operator in momentum space. In this work, we introduce an alternative approach to topology which is based on the scattering matrix of a quantum walk, adapting concepts from time-independent systems. For quantum walks with gaps in the quasienergy spectrum at 0 and $\ensuremath{\pi}$, we find three different types of topological invariants, which apply dependent on the symmetries of the system. These determine the number of protected boundary states at an interface between two quantum-walk regions. Quantum walks with an unequal number of leftward and rightward shifts per cycle are characterized by the number of perfectly transmitting unidirectional modes they support, which is equal to their nontrivial quasienergy winding. Our classification provides a unified framework that includes all known types of topology in one-dimensional discrete-time quantum walks and is very well suited for the analysis of finite-size and disorder effects. We provide a simple scheme to directly measure the topological invariants in an optical quantum-walk experiment.