A local test for global extrema in the dispersion relation of a periodic graph
arXiv:2004.12931 · 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 its global extremum if the dimension of the periodicity group is three 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.
arXiv comment30 pages, 7 figures; incorporated corrections from Lior Alon and anonymous referees; expanded explanation of the proof ideas in the introduction
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