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The Witten index for 1D supersymmetric quantum walks with anisotropic coins

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Abstract

Chiral symmetric discrete-time quantum walks possess supersymmetry, and their associated Witten indices can be naturally defined. The Witten index is known to give a lower bound for the number of topologically protected bound states. The purpose of this paper is to give a complete classification of the Witten index for a one-dimensional split-step quantum walk. It turns out that the Witten index of this model exhibits a striking similarity to that of a Dirac particle model in supersymmetric quantum mechanics.

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Notes

  1. That is to say, \(\pm 1\) are not accumulation points of the spectrum of U.

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Acknowledgements

The authors are deeply indebted to the members of the Shinshu Mathematical Physics Group for extremely valuable discussions and comments. A. S. was supported by JSPS KAKENHI Grant Number JP18K03327. This work was also supported by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.

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Authors and Affiliations

  1. Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, Wakasato, Nagano, 380-8553, Japan

    Akito Suzuki

  2. School of Computer Science, Engineering and Mathematics, Flinders University, 1284 South Road, Clovelly Park, SA, 5042, Australia

    Yohei Tanaka

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  1. Akito Suzuki
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Correspondence to Akito Suzuki.

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Suzuki, A., Tanaka, Y. The Witten index for 1D supersymmetric quantum walks with anisotropic coins. Quantum Inf Process 18, 377 (2019). https://doi.org/10.1007/s11128-019-2485-1

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