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Discrete Bethe-Sommerfeld conjecture for triangular, square, and hexagonal lattices

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Abstract

We prove the discrete Bethe-Sommerfeld conjecture on the graphene lattice, on its dual lattice (the triangular lattice), and on the extended Harper lattice. For each of these lattice geometries, we analyze the behavior of small periodic potentials. In particular, we provide sharp bounds on the number of gaps that may perturbatively open, we describe sharp arithmetic criteria on the periods that ensure that no gaps open, and we characterize those energies at which gaps may open in the perturbative regime. In all three cases, we provide examples that open the maximal number of gaps and estimate the scaling behavior of the gap lengths as the coupling constant goes to zero.

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Acknowledgements

We would like to thank Svetlana Jitomirskaya for comments on an earlier version of the manuscript, and Tom Spencer for useful discussions. R. H. would like to thank IAS, Princeton, for its hospitality during the 2017–18 academic year, and Virginia Tech for its hospitality during which part of the work was done. R. H. is supported in part by the National Science Foundation under Grant No. DMS-1638352 and DMS-1800689. J. F. was supported in part by an AMS Simons Travel Grant 2016–2018 and Simons Collaboration Grant #711663.

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Authors and Affiliations

  1. Department of Mathematics, MCS 470, Texas State University, 601 University Dr., San Marcos, TX, 78666, USA

    Jake Fillman

  2. Department of Mathematics, Louisiana State University, Lockett Hall, Baton Rouge, LA, 70802, USA

    Rui Han

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  1. Jake Fillman
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  2. Rui Han
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Correspondence to Jake Fillman or Rui Han.

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Fillman, J., Han, R. Discrete Bethe-Sommerfeld conjecture for triangular, square, and hexagonal lattices. JAMA 142, 271–321 (2020). https://doi.org/10.1007/s11854-020-0138-z

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  • DOI: https://doi.org/10.1007/s11854-020-0138-z

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