Non-Compact 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 C2 rotational symmetry, we prove that the complement of fragile topological bands cannot be compact, even if it is an atomic insulator. Third, for C4 symmetry, our findings imply that there exist fragile bands with zero correlation length. Fourth, for a C3-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.

Superfluid weight bounds from symmetry and quantum geometry in flat bands

“…In this case the total Chern number is zero. However, the RSIs are δ 1a = δ 1b = δ 1c = −1 and δ 1d = 1, so our bound is nonzero: [49] as their conduction bands, we learn that an OWC can have just as strong a lower bound as a fragile topological insulator.…”

“…(F8) to evaluate traces, we find With these constraints, we use the concentration lemma to bound The results of this Appendix show that nonzero lower bounds can be obtained in the case of generic orbital positions, but are suboptimal and can be improved with future work. The study of optimal bounds in the generic case is enriched by the existence of non-compact obstructed atomic insulators [49] and may be a fruitful area of study.…”

“…From a materials perspective, although topological bands are abundant within real crystals, a significant portion are topologically trivial at the Fermi level. Topo-logically trivial bands have a finer classification which divides them into trivial atomic bands, where electrons are exponentially localized at the atomic sites, and bands with obstructed Wannier centers (OWCs) which, while exponentially localized, are necessarily centered off the atoms [47][48][49] . Remarkably, we show that, like topological bands, OWCs too have a nonzero, lower-bounded Fubini-Study metric despite the absence of Berry curvature 50,51 .…”

“…so the mean-field superfluid weight in Eq. ( 1) is nonzero despite the model being topologically trivial and having compact Wannier functions (zero correlation length) 49 .…”

“…( 14) in terms of the correlation function shows that long-ranged Wannier functions are not essential to the lower bound. Any Wannier function which is supported over multiple unit cells 36,49 , as can be enforced by symmetry in a OWC state, produces a quantized RSI lower bound. Our derivation is general for arbitrary bands and arbitrary symmetries.…”