Non-Compact Atomic Insulators

TL;DR

This paper investigates the conditions under which atomic insulators can have strictly compact Wannier states, revealing topological and symmetry-related obstructions to their existence and providing new insights into Wannier state localization.

Contribution

It establishes criteria for compact Wannier states, analyzes symmetry effects on their existence, and conjectures universal obstructions for finite Wannier sizes in atomic insulators.

Findings

Compact Wannier states may be obstructed even in topologically trivial insulators.

Symmetry considerations (C2, C3, C4) influence the possibility of compact Wannier states.

There is a conjectured universal obstruction to finite Wannier sizes in atomic insulators.

Abstract

We study the conditions for Bloch bands to be spanned by symmetric and strictly compact Wannier states that have zero overlap with all lattice sites beyond a certain range. Similar to the characterization of topological insulators in terms of an algebraic (rather than exponential) localization of Wannier states, we find that there may be impediments to the compact localization even of topologically "trivial" obstructed atomic insulators. These insulators admit exponentially-localized Wannier states centered at unoccupied orbitals of the crystalline lattice. First, we establish a sufficient condition for an insulator to have a compact representative. Second, for $\mathcal{C}_2$ rotational symmetry, we prove that the complement of fragile topological bands cannot be compact, even if it is an atomic insulator. Third, for $\mathcal{C}_4$ symmetry, our findings imply that there exist fragile bands with zero correlation length. Fourth, for a $\mathcal{C}_3$-symmetric atomic insulator, we explicitly derive that there are no compact Wannier states overlapping with less than $18$ lattice sites. We conjecture that this obstruction generalizes to all finite Wannier sizes. Our results can be regarded as the stepping stone to a generalized theory of Wannier states beyond dipole or quadrupole polarization.