Compact Wannier Functions in One Dimension

TL;DR

This work answers when compact Wannier functions can exist in one dimension by proving equivalence between the existence of $CWF$s and the strict locality of the band projector ($P$). It provides a complete constructive method using Laurent-polynomial unitary rotations to transform $P(k)$ into a diagonal form, yielding $CWF$s with support limited to $R+1$ cells and establishing their uniqueness for single-band cases. The study clarifies the relationship between $CWF$s, MLWFs, and compact Wannier-type functions (CWTs), showing that MLWFs are generally not compact, though NN projectors do yield compact MLWFs and generalized Wannier functions in non-LTI settings. It further develops NN-specific results, giving explicit procedures for both NN without translational invariance and NN with translational invariance, and extends to larger hopping ranges via a supercell strategy. The findings have implications for flat-band models, topological no-go theorems in 1D, and potential extensions to higher dimensions and interacting systems through Green’s-function language.

Abstract

Wannier functions have widespread utility in condensed matter physics and beyond. Topological physics, on the other hand, has largely involved the related notion of compactly-supported Wannier-type functions, which arise naturally in flat bands. In this paper, we establish a connection between these two notions, by finding the necessary and sufficient conditions under which compact Wannier functions exist in one dimension. We present an exhaustive construction of models with compact Wannier functions and show that the Wannier functions are unique, and in general, distinct from the corresponding maximally-localized Wannier functions.