The Topological Classification of One-Dimensional Symmetric QuantumWalks

authored by: Christopher Cedzich, T. Geib, F. A. Grünbaum, C. Stahl, L. Velázquez, A. H. Werner, R. F. Werner
Abstract: We give a topological classification of quantum walks on an infinite 1D lattice, which obey one of the discrete symmetry groups of the tenfold way, have a gap around some eigenvalues at symmetry protected points, and satisfy a mild locality condition. No translation invariance is assumed. The classification is parameterized by three indices, taking values in a group, which is either trivial, the group of integers, or the group of integers modulo 2, depending on the type of symmetry. The classification is complete in the sense that two walks have the same indices if and only if they can be connected by a norm-continuous path along which all the mentioned properties remain valid. Of the three indices, two are related to the asymptotic behavior far to the right and far to the left, respectively. These are also stable under compact perturbations. The third index is sensitive to those compact perturbations which cannot be contracted to a trivial one. The results apply to the Hamiltonian case as well. In this case, all compact perturbations can be contracted, so the third index is not defined. Our classification extends the one known in the translation- invariant case, where the asymptotic right and left indices add up to zero, and the third one vanishes, leaving effectively only one independent index. When two translation-invariant bulks with distinct indices are joined, the left and right asymptotic indices of the joined walk are thereby fixed, and there must be eigenvalues at 1 or -1 (bulk-boundary correspondence). Their location is governed by the third index. We also discuss how the theory applies to finite lattices, with suitable homogeneity assumptions.
Organisation(s): Institute of Theoretical Physics
CRC 1227 Designed Quantum States of Matter (DQ-mat)
External Organisation(s): University of California at Berkeley
Universidad de Zaragoza
University of Copenhagen
Type: Article
Journal: Annales Henri Poincaré
Volume: 19
Pages: 325–383
No. of pages: 59
ISSN: 1424-0637
Publication date: 02.2018
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Statistical and Nonlinear Physics, Nuclear and High Energy Physics, Mathematical Physics
Electronic version(s): https://doi.org/10.48550/arXiv.1611.04439 (Access: Open)
https://doi.org/10.1007/s00023-017-0630-x (Access: Closed)

BibTeX

@article{d1716204e6b74ecb96ee4d1fb87bd504,
title = "The Topological Classification of One-Dimensional Symmetric QuantumWalks",
abstract = "We give a topological classification of quantum walks on an infinite 1D lattice, which obey one of the discrete symmetry groups of the tenfold way, have a gap around some eigenvalues at symmetry protected points, and satisfy a mild locality condition. No translation invariance is assumed. The classification is parameterized by three indices, taking values in a group, which is either trivial, the group of integers, or the group of integers modulo 2, depending on the type of symmetry. The classification is complete in the sense that two walks have the same indices if and only if they can be connected by a norm-continuous path along which all the mentioned properties remain valid. Of the three indices, two are related to the asymptotic behavior far to the right and far to the left, respectively. These are also stable under compact perturbations. The third index is sensitive to those compact perturbations which cannot be contracted to a trivial one. The results apply to the Hamiltonian case as well. In this case, all compact perturbations can be contracted, so the third index is not defined. Our classification extends the one known in the translation- invariant case, where the asymptotic right and left indices add up to zero, and the third one vanishes, leaving effectively only one independent index. When two translation-invariant bulks with distinct indices are joined, the left and right asymptotic indices of the joined walk are thereby fixed, and there must be eigenvalues at 1 or -1 (bulk-boundary correspondence). Their location is governed by the third index. We also discuss how the theory applies to finite lattices, with suitable homogeneity assumptions.",
author = "Christopher Cedzich and T. Geib and Gr{\"u}nbaum, {F. A.} and C. Stahl and L. Vel{\'a}zquez and Werner, {A. H.} and Werner, {R. F.}",
note = "Funding information: C. Cedzich, T. Geib, C. Stahl, and R. F. Werner acknowledge support from the ERC grant DQSIM, the DFG SFB 1227 DQmat, and the European project SIQS. The work of L. Vel{\'a}zquez is partially supported by the research project MTM2014-53963-P from the Ministry of Science and Innovation of Spain and the European Regional Development Fund (ERDF) and by Project E-64 of Diputaci{\'o}n General de Arag{\'o}n (Spain). A. H. Werner thanks the Humboldt Foundation for its support with a Feodor Lynen Fellowship and the VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059).",
year = "2018",
month = feb,
doi = "10.48550/arXiv.1611.04439",
language = "English",
volume = "19",
pages = "325–383",
journal = "Annales Henri Poincar{\'e}",
issn = "1424-0637",
publisher = "Birkhauser Verlag Basel",
number = "2",
}