TY - JOUR

T1 - Quantum Walks

T2 - Schur Functions Meet Symmetry Protected Topological Phases

AU - Cedzich, C.

AU - Geib, T.

AU - Grünbaum, F. A.

AU - Velázquez, L.

AU - Werner, A. H.

AU - Werner, R. F.

N1 - Funding information: C. Cedzich acknowledges support by the Excellence Initiative of the German Federal and State Governments (Grant ZUK 81) and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) (Project B01 of CRC 183 and Grant No. 441423094). T. Geib 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. Velázquez in this publication is part of the I+D+i Project MTM2017-89941-P funded by MCIN/ AEI/10.13039/501100011033/ and ERDF “Una manera de hacer Europa”, the Project UAL18-FQM-B025-A (UAL/CECEU/FEDER) and the Projects E26_17 and E48_20R from Diputación General de Aragón (Spain) and ERDF “Construyendo Europa desde Aragón”. A. H. Werner thanks the Alexander von Humboldt Foundation for its support with a Feodor Lynen Fellowship and the VILLUM FONDEN for its support via a Villum Young Investigator Grant (Grant No. 25452) and the QMATH Centre of Excellence (Grant No. 10059) and acknowledges financial support from the European Research Council (ERC Grant Agreement No. 337603). C.?Cedzich acknowledges support by the Excellence Initiative of the German Federal and State Governments (Grant ZUK 81) and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) (Project B01 of CRC 183 and Grant No. 441423094). T.?Geib 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?zquez in this publication is part of the I+D+i Project MTM2017-89941-P funded by MCIN/ AEI/10.13039/501100011033/ and ERDF ?Una manera de hacer Europa?, the Project UAL18-FQM-B025-A (UAL/CECEU/FEDER) and the Projects E26_17 and E48_20R from Diputaci?n General de Arag?n (Spain) and ERDF ?Construyendo Europa desde Arag?n?. A. H. Werner thanks the Alexander von Humboldt Foundation for its support with a Feodor Lynen Fellowship and the VILLUM FONDEN for its support via a Villum Young Investigator Grant (Grant No. 25452) and the QMATH Centre of Excellence (Grant No. 10059) and acknowledges financial support from the European Research Council (ERC Grant Agreement No. 337603). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

PY - 2022/1

Y1 - 2022/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85122057478&partnerID=8YFLogxK

U2 - 10.1007/s00220-021-04284-8

DO - 10.1007/s00220-021-04284-8

M3 - Article

VL - 389

SP - 31

EP - 74

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

ER -