Existence and computation of generalized Wannier functions for nonperiodic systems in two dimensions and higher
Abstract
Exponentiallylocalized Wannier functions (ELWFs) are an orthonormal basis of the Fermi projection of a material consisting of functions which decay exponentially fast away from their maxima. When the material is insulating and crystalline, conditions which guarantee existence of ELWFs in dimensions one, two, and three are wellknown, and methods for constructing the ELWFs numerically are welldeveloped. We consider the case where the material is insulating but not necessarily crystalline, where much less is known. In one spatial dimension, Kivelson and NenciuNenciu have proved ELWFs can be constructed as the eigenfunctions of a selfadjoint operator acting on the Fermi projection. In this work, we identify an assumption under which we can generalize the KivelsonNenciuNenciu result to two dimensions and higher. Under this assumption, we prove that ELWFs can be constructed as the eigenfunctions of a sequence of selfadjoint operators acting on the Fermi projection. We conjecture that the assumption we make is equivalent to vanishing of topological obstructions to the existence of ELWFs in the special case where the material is crystalline. We numerically verify that our construction yields ELWFs in various cases where our assumption holds and provide numerical evidence for our conjecture.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 arXiv:
 arXiv:2003.06676
 Bibcode:
 2020arXiv200306676L
 Keywords:

 Mathematical Physics;
 Condensed Matter  Mesoscale and Nanoscale Physics;
 Mathematics  Numerical Analysis;
 Physics  Computational Physics
 EPrint:
 64 pages, 15 figures. Many revisions: (1) Proof generalized WFs we construct are (a) WFs when system periodic (b) respectful of timereversal, so gap conditions imply topological triviality (2) Proofs streamlined using integral kernel of P