Honeycomb Schrödinger Operators in the Strong Binding Regime

Charles L. Fefferman, James P. Lee-Thorp, Michael I. Weinstein

Research output: Contribution to journalArticlepeer-review

54 Scopus citations


In this article, we study the Schrödinger operator for a large class of periodic potentials with the symmetry of a hexagonal tiling of the plane. The potentials we consider are superpositions of localized potential wells, centered on the vertices of a regular honeycomb structure corresponding to the single electron model of graphene and its artificial analogues. We consider this Schrödinger operator in the regime of strong binding, where the depth of the potential wells is large. Our main result is that for sufficiently deep potentials, the lowest two Floquet-Bloch dispersion surfaces, when appropriately rescaled, converge uniformly to those of the two-band tight-binding model (Wallace, 1947 [56]). Furthermore, we establish as corollaries, in the regime of strong binding, results on (a) the existence of spectral gaps for honeycomb potentials that break PT symmetry and (b) the existence of topologically protected edge states—states that propagate parallel to and are localized transverse to a line defect or “edge”—for a large class of rational edges, and that are robust to a class of large transverse-localized perturbations of the edge. We believe that the ideas of this article may be applicable in other settings for which a tight-binding model emerges in an extreme parameter limit.

Original languageEnglish (US)
Pages (from-to)1178-1270
Number of pages93
JournalCommunications on Pure and Applied Mathematics
Issue number6
StatePublished - Jun 2018

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


Dive into the research topics of 'Honeycomb Schrödinger Operators in the Strong Binding Regime'. Together they form a unique fingerprint.

Cite this