Algebraic localization implies exponential localization in non-periodic insulators

TL;DR

This paper proves that in non-periodic insulators, the existence of a basis with finite fifth moment implies the existence of exponentially localized Wannier functions, supporting a conjecture about localization properties.

Contribution

It establishes a link between algebraic localization (finite moments) and exponential localization for non-periodic insulators in two dimensions.

Findings

Finite fifth moment basis implies exponential localization

Supports the Localization Dichotomy Conjecture for non-periodic systems

Extends known results from periodic to non-periodic insulators

Abstract

Exponentially-localized 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 exponentially-localized 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 non-periodic 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 non-periodic systems recently proposed by Marcelli, Monaco, Moscolari, and Panati