Identifying Instabilities with Quantum Geometry in Flat-Band Systems

The absence of a well-defined Fermi surface in flat-band systems challenges the conventional understanding of instabilities toward Landau order based on nesting. We investigate the existence of an intrinsic nesting structure encoded in the band geometry [i.e., the wave functions of the flat band(s)], which leads to a maximal susceptibility at the mean-field level and, thus, determines the instability toward ordered phases. More generally, we show that, for a given band structure and observable, we can define two vector fields: one which corresponds to the Bloch vector of the projection operator onto the manifold of flat bands and another which is "dressed" by the observable. The overlap between the two vector fields, possibly shifted by a momentum vector Q, fully determines the mean-field susceptibility of the corresponding order parameter. When the overlap is maximized, so is the susceptibility, and this geometrically corresponds to "perfect nesting" of the band structure. In that case, we show that the correlation length of this order parameter, even for Q≠0, is entirely characterized by a generalized quantum metric in an intuitive manner and is, therefore, lower bounded in topologically nontrivial bands. As an example, we demonstrate hidden nesting for staggered antiferromagnetic spin order in an exactly flat-band model, which is notably different from the general intuition that flat bands are closely associated with ferromagnetism. We check the actual emergence of this long-range order using the determinantal quantum Monte Carlo algorithm. Additionally, we demonstrate that a Fulde-Ferrell-Larkin-Ovchinnikov-like state (pairing with nonzero center of mass momentum) can arise in flat bands upon breaking time-reversal symmetry, even if Zeeman splitting is absent.