Quantum Walks: Schur Functions Meet Symmetry Protected Topological Phases

• DOI: 10.1007/s00220-021-04284-8
• Publication Date: 2021-12-29T12:02:40Z
• Authors: C. Cedzich; T. Geib; F. A. Grünbaum; L. Velázquez; A. H. Werner; R. F. Werner
• Keywords: Quantum walk; Schur algebra

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