Maximally Localized Wannier Functions for Strictly Local Projectors

The localization properties of band projectors and their associated Wannier functions are closely related. In particular, a band projector in a 1d lattice is strictly local (SL) if and only if there exists an associated Wannier basis consisting of compactly supported wavefunctions [1]. We improve the bound on the degree of localization of Wannier functions associated with SL projectors. Specifically, we show that for an SL projector with a maximum hopping distance of b, it is always possible to obtain compactly supported Wannier functions with spatial support of at most 2b cells. Additionally, these wavefunctions are the corresponding maximally localized (generalized) Wannier functions in the presence (absence) of translational invariance, within a size b supercell representation of the lattice. Based on this result, we propose a method for the construction of arbitrary SL projectors in 1d, and demonstrate its potential for constructing model flat-band Hamiltonians in 1d.

[1] Sathe, P., Harper, F., & Roy, R. (2020) arXiv:2008.05528.