Abstract
Exponentially-localized 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 well-known, and methods for constructing ELWFs numerically are well-developed. We consider the case where the material is insulating but not necessarily crystalline, where much less is known. In one spatial dimension, Kivelson and Nenciu-Nenciu have proved ELWFs can be constructed as the eigenfunctions of a self-adjoint operator acting on the Fermi projection. In this work, we identify an assumption under which we can generalize the Kivelson–Nenciu–Nenciu result to two dimensions and higher. Under this assumption, we prove that ELWFs can be constructed as the eigenfunctions of a sequence of self-adjoint operators acting on the Fermi projection.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1269-1323 |
| Number of pages | 55 |
| Journal | Archive For Rational Mechanics And Analysis |
| Volume | 243 |
| Issue number | 3 |
| DOIs | |
| State | Published - Mar 2022 |
| Externally published | Yes |
Bibliographical note
Funding Information:This work is supported in part by the National Science Foundation via Grant DMS-1454939 and the Department of Energy via Grant DE-SC0019449. K.D.S. is also supported in part by a National Science Foundation Graduate Research Fellowship under Grant No. DGE-1644868.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE, part of Springer Nature.