A novel strong-coupling expansion for flat band electrons with concentrated dispersion

Describing electron correlations in two-dimensional systems with flat, isolated bands is notoriously challenging due to the absence of a small parameter controlling interaction strength. We introduce a novel strong-coupling perturbative framework — the concentrated dispersion expansion — tailored for bands that are dispersive only within a small region of size s<<1 near the Γ-point, and nearly flat elsewhere in the Brillouin zone. This situation arises naturally in Chern bands of moiré materials, where the small parameter “s” emerges from a concentrated Berry curvature. In real space, such dispersion corresponds to single-particle or correlated hopping terms with amplitudes that decay over a length scale 1/s, and scale in magnitude as s^2. In the limit s<<1, the theory becomes perturbatively controlled: long, self-avoiding tunneling paths dominate due to the balance between small amplitudes and large phase-space volume, while paths revisiting the same site are suppressed. We illustrate our general approach using a modified Hubbard model with exponentially decaying hopping. At first order in s^2, it already captures nontrivial effects, such as intrinsic spectral broadening induced by fluctuating local moments and the associated resistivity — features that are hard to access using conventional perturbative schemes in a controlled manner. Our approach provides a systematic method for studying correlated phases in which Mott physics and itinerant degrees of freedom coexist within a single band.