Symmetries, topological phases, and bound states in the one-dimensional quantum walk

By J. Asbóth. 2012

161 citations recorded by the source corpus.

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 Z2 \times Z2 topological invariant in a simple way that is consistent with a general definition for 1-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 correspondance. 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.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/1d510c56a6ce94d62e2b29443c63ed8178e9be72

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Nathan Goldman
2. A. Amo
3. I. Carusotto
4. David I. Schuster
5. H. Price
6. T. Ozawa
7. Michael J. Hartmann
8. Steven M. Girvin

In Atomic and Molecular Physics, and Optics

1. Nathan Goldman
2. A. Amo
3. T. Ozawa
4. H. Price
5. M. Rechtsman
6. I. Carusotto
7. David I. Schuster
8. Jonathan Simon

Citations in the corpus

• Exceptional topology of non-Hermitian systems (2019, unknown): “…made topologically non-trivial , resulting in the appearance of topological phases in quantum walks as predicted in theory (Kitagawa et al. , 2010; Asbóth, 2012) and as shown experimentally in discrete-time quantum walks (Kitagawa et al. , 2012; Cardano et al. , 2016; Barkhofen et al. , 2017;…”
• Topological photonics (2018, background): “Such topological invariants (winding numbers) were identified by Kitagawa, Berg et al. (2010), Asbóth (2012), Rudner et al. (2013), Carpentier et al. (2015), Nathan and Rudner (2015), Bi et al. (2017b), and Yao, Yan, and Wang (2017) and were indeed shown to depend on the complete time-evolution…”