Compact Wannier functions in one dimension

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.