Stable Topology in Exactly Flat Bands

Abstract

Topological flat bands (FBs) offer an ideal platform for realizing exotic topological phases, such as fractional Chern insulators, yet their realization with both exact flatness and stable topology in local lattice models has been long hindered by fundamental no-go theorems. The obstruction to topological FBs is also manifested as the absence of exact Gaussian tensor-network state (TNS) representations for topological insulators and superconductors. Here, we overcome this barrier by demonstrating the existence of critical topological FBs (CTFBs) in finite-range hopping models. They saturate the no-go theorems via a unique structure of Bloch wavefunctions: While continuous over the whole Brillouin zone, the projector $P(\mathbf{k})$ onto FBs are non-analytic at isolated band touching points, thereby relaxing the inherent restrictions on the coexistence of exact flatness and stable topology. We establish a general principle to construct CTFBs, as well as their parent Hamiltonians, that carry desired topological invariants in given space groups. Explicit examples exhibiting Chern numbers 1, 2, 3 in 2D, strong $\mathbb{Z}_2$ index in 2D, and strong $\mathbb{Z}_2$ index in 3D are provided. Furthermore, an automated algorithm identifies more than 50,000 symmetry-indicated CTFBs. Achieved without fine-tuning, these FBs host nontrivial topology that is robust against arbitrary symmetry-preserving perturbations such as gap-opening terms. Filling such CTFBs yields short-range entangled topological states that exhibit power-law correlations. Crucially, all filled CTFB states admit exact TNS representations with finite bond dimensions, providing a tractable starting point for exploring strongly correlated topological matter.