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A local test for global extrema in the dispersion relation of a periodic graph

Gregory Berkolaiko, Yaiza Canzani, Graham Cox and Jeremy Louis Marzuola

Vol. 4 (2022), No. 2, 257–286
DOI: 10.2140/paa.2022.4.257
Abstract

We consider a family of periodic tight-binding models (combinatorial graphs) that have the minimal number of links between copies of the fundamental domain. For this family we establish a local condition of second derivative type under which the critical points of the dispersion relation can be recognized as global maxima or minima. Under the additional assumption of time-reversal symmetry, we show that any local extremum of a dispersion band is in fact a global extremum if the dimension of the periodicity group is 3 or less, or (in any dimension) if the critical point in question is a symmetry point of the Floquet–Bloch family with respect to complex conjugation. We demonstrate that our results are nearly optimal with a number of examples.

Keywords
dispersion relation, tight-binding model, graph Laplacian, Floquet–Bloch, band gaps
Mathematical Subject Classification
Primary: 35Q40, 81Q10, 81Q35
Milestones
Received: 29 March 2021
Revised: 13 December 2021
Accepted: 26 January 2022
Published: 16 October 2022
Authors
Gregory Berkolaiko
Department of Mathematics
Texas A&M University
College Station, TX
United States
Yaiza Canzani
Department of Mathematics
University of North Carolina
Chapel Hill, NC
United States
Graham Cox
Department of Mathematics and Statistics
Memorial University of Newfoundland
St. John’s, NL
Canada
Jeremy Louis Marzuola
Department of Mathematics
University of North Carolina
Chapel Hill, NC
United States