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Abstract
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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.
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Keywords
dispersion relation, tight-binding model, graph Laplacian,
Floquet–Bloch, band gaps
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Mathematical Subject Classification
Primary: 35Q40, 81Q10, 81Q35
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Milestones
Received: 29 March 2021
Revised: 13 December 2021
Accepted: 26 January 2022
Published: 16 October 2022
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