# Compact Wannier Functions in One Dimension

## TL;DR

This paper explores the conditions for the existence of compact Wannier functions in one-dimensional systems, providing a comprehensive construction of models and highlighting their uniqueness and differences from maximally-localized Wannier functions.

## Contribution

It establishes necessary and sufficient conditions for compact Wannier functions in 1D and offers an exhaustive construction of such models, revealing their uniqueness.

## Key findings

- Conditions for existence of compact Wannier functions in 1D
- Construction of models with compact Wannier functions
- Compact Wannier functions are generally distinct from maximally-localized ones

## 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.

---
Source: https://tomesphere.com/paper/2302.11608