Quantum Walks

Schur Functions Meet Symmetry Protected Topological Phases

authored by
C. Cedzich, T. Geib, F. A. Grünbaum, L. Velázquez, A. H. Werner, R. F. Werner
Abstract

This paper uncovers and exploits a link between a central object in harmonic analysis, the so-called Schur functions, and the very hot topic of symmetry protected topological phases of quantum matter. This connection is found in the setting of quantum walks, i.e. quantum analogs of classical random walks. We prove that topological indices classifying symmetry protected topological phases of quantum walks are encoded by matrix Schur functions built out of the walk. This main result of the paper reduces the calculation of these topological indices to a linear algebra problem: calculating symmetry indices of finite-dimensional unitaries obtained by evaluating such matrix Schur functions at the symmetry protected points ±1. The Schur representation fully covers the complete set of symmetry indices for 1D quantum walks with a group of symmetries realizing any of the symmetry types of the tenfold way. The main advantage of the Schur approach is its validity in the absence of translation invariance, which allows us to go beyond standard Fourier methods, leading to the complete classification of non-translation invariant phases for typical examples.

Organisation(s)
Institute of Theoretical Physics
CRC 1227 Designed Quantum States of Matter (DQ-mat)
External Organisation(s)
University Hospital Düsseldorf
University of California at Berkeley
Universidad de Zaragoza
University of Copenhagen
Type
Article
Journal
Communications in Mathematical Physics
Volume
389
Pages
31-74
No. of pages
44
ISSN
0010-3616
Publication date
01.2022
Publication status
Published
Peer reviewed
Yes
ASJC Scopus subject areas
Statistical and Nonlinear Physics, Mathematical Physics
Electronic version(s)
https://arxiv.org/abs/1903.07494 (Access: Open)
https://doi.org/10.1007/s00220-021-04284-8 (Access: Open)