Speaker
Description
Topological flat bands (FBs) offer an ideal platform for realizing exotic topological phases and exploringquantum geometric effects, yet their realization with both exact flatness and stable topology in local latticemodels has been long hindered by fundamental no-go theorems. The obstruction to topological FBs is alsomanifested as the absence of exact Gaussian tensor-network state (TNS) representations for topological insula-tors and superconductors. Here, we overcome this barrier by demonstrating the existence of critical topologicalFBs (CTFBs) in finite-range hopping models. They saturate the no-go theorems via a unique structure of Blochwavefunctions: While continuous over the whole Brillouin zone, the projector P(k) onto FBs is non-analyticat isolated band touching points, thereby relaxing the inherent restrictions on the coexistence of exact flatnessand stable topology. Filling such CTFBs yields short-range entangled topological states that exhibit power-lawcorrelations due to the non-analyticity. We then establish a general symmetry-based principle to systemati-cally construct CTFBs, as well as their parent Hamiltonians, that carry desired topological invariants in givenspace groups. It utilizes the bipartite structures and requires no further fine-tuning, and the topology is robustagainst arbitrary symmetry-preserving perturbations that violate the bipartite condition and gap the touchingpoints. Remarkably, independent tuning of the local quantum geometry while maintaining flatness and topology is allowed. Examples exhibiting Chern numbers 1, 2, 3, 6, in 2D, and strong Z2 indices in 2D and 3D aredemonstrated for concreteness. An automated algorithm further identifies more than 50,000 symmetry-indicatedCTFBs, including crystalline and higher-order topological ones. In the end, we show that the bipartite structurenaturally endows the filled CTFB states with exact TNS representations with finite bond dimensions. By bridging the topological band theory to TNS methods, CTFB provides a novel tractable starting point for exploringstrongly correlated topological matter, with potential relevance to realistic material realizations.