New class of exactly flat topological bands - compact localised states protected by local graph topology

Abstract:Strongly correlated quantum matter is fundamentally defined by the tension between non-commuting quantum operators. Hamiltonians exhibiting macroscopic degeneracies are of general interest in this field because they imply an infinite susceptibility to any non-commuting perturbation. In moiré heterostructures, engineering such extensive degeneracies in the kinetic Hamiltonian creates a fertile garden for exotic strongly correlated phases of matter to emerge from the resulting flat bands. Here, we introduce a prescription to construct an infinite family of exact flat band Hamiltonians supported on the faces of arbitrary graphs. We demonstrate this algorithm on the faces of four Bravais lattices. Using a discrete graph generalization of the Atiyah-Singer index theorem, we prove that the extensive degeneracies of such face-graph Hamiltonians are protected by the local topology of the face-graph connectivity. The resulting macroscopic null spaces yield compact localised states that remain localised over time due to frustration in hopping pathways. We discuss the broad implications of such non-dispersing quantum modes in diverse settings, from arrested dynamics in quantum networks and quantum machine learning algorithms to Majorana-free topological quantum computation.

19 pages main text, 8 figures, 1 appendix. Comments welcome and will be gratefully acknowledged upon submission

Strongly Correlated Electrons (cond-mat.str-el); Disordered Systems and Neural Networks (cond-mat.dis-nn); Materials Science (cond-mat.mtrl-sci); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

arXiv:2607.22831 [cond-mat.str-el]

(or arXiv:2607.22831v1 [cond-mat.str-el] for this version)

https://doi.org/10.48550/arXiv.2607.22831 arXiv-issued DOI via DataCite (pending registration)