General Construction from Topological Loop States to Topological Invariants in Exactly Flat Bands

Electronic flat bands have localized Wannier-like orbitals as zero modes. In the Lieb or kagome models, the localized orbitals satisfy a topological condition that entails two noncontractible loop eigenstates along the x or y axis in real space and one topological band touching point with other bands in momentum space. In these topological flat bands, the Bloch state at the touching point is ill defined and so is any topological invariant for the entire band. We propose a new topological condition in which the loop states in different directions are linearly dependent. Its satisfaction removes the singularity at the band touching point and enforces nontrivial, well-defined topological invariants. Enforcing the new condition, we obtain topological-topological flat (top^{2}-flat) bands in 2D and 3D with nontrivial invariants including the Chern number, the Z_{2} invariants, and the topological-crystalline invariants. Under small, generic interactions, top^{2}-flat bands flow to correlated topological insulators with a symmetric mass term, and specially designed interacting models can have top^{2}-flat bands as exact zero modes.