Identifying Instabilities with Quantum Geometry in Flat Band Systems

Flat-band systems lack a well-defined Fermi surface, challenging the conventional nesting-based picture of instabilities toward Landau order. We introduce generalized Bloch vectors to reveal an intrinsic nesting structure encoded in the band geometry, leading to maximal susceptibility at the mean-field level and driving ordered phases. We further show that the correlation length of an order parameter satisfying perfect nesting—even at finite momentum—is entirely characterized by a generalized quantum metric, and is therefore lower-bounded in topologically non-trivial bands. As an example, we demonstrate hidden nesting for staggered antiferromagnetic spin order in an exactly flat-band model—contrary to the usual association of flat bands with ferromagnetism. Determinantal quantum Monte Carlo confirms the emergence of this long-range order. We further show that breaking time-reversal symmetry can induce an FFLO-like state without Zeeman splitting.