Algebraic localization implies exponential localization in nonperiodic insulators
Abstract
Exponentiallylocalized Wannier functions are a basis of the Fermi projection of a Hamiltonian consisting of functions which decay exponentially fast in space. In two and three spatial dimensions, it is well understood for periodic insulators that exponentiallylocalized Wannier functions exist if and only if there exists an orthonormal basis for the Fermi projection with finite second moment (i.e. all basis elements satisfy $\int \boldsymbol{x}^2 w(\boldsymbol{x})^2 \,\text{d}{\boldsymbol{x}} < \infty$). In this work, we establish a similar result for nonperiodic insulators in two spatial dimensions. In particular, we prove that if there exists an orthonormal basis for the Fermi projection which satisfies $\int \boldsymbol{x}^{5 + \epsilon} w(\boldsymbol{x})^2 \,\text{d}{\boldsymbol{x}} < \infty$ for some $\epsilon > 0$ then there also exists an orthonormal basis for the Fermi projection which decays exponentially fast in space. This result lends support to the Localization Dichotomy Conjecture for nonperiodic systems recently proposed by Marcelli, Monaco, Moscolari, and Panati
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 arXiv:
 arXiv:2101.02626
 Bibcode:
 2021arXiv210102626L
 Keywords:

 Mathematical Physics;
 Condensed Matter  Mesoscale and Nanoscale Physics
 EPrint:
 32 pages. Simplified and streamlined proofs using updated results from arxiv:2003.06676v5