Band-minimum degeneracy is not enough: density-of-states control of low-density ferromagnetism

Abstract: Recent ultracold-atom experiments have observed ferromagnetic correlations in a geometrically tunable Fermi--Hubbard lattice at intermediate densities. Motivated by subsequent theoretical work that connected the low-density limit of the same model to the Mueller--Hartmann mechanism for itinerant ferromagnetism, we investigate the stability of ferromagnetism across a broad density range using the T-matrix approximation, a controlled approach in the dilute limit. We reproduce the ferromagnetic regime observed experimentally at intermediate densities and systematically compare finite-size and thermodynamic-limit calculations in the dilute regime. We find that the low-density ferromagnetic phase reported for the one-diagonal hopping lattice is strongly suppressed with increasing system size and disappears in the thermodynamic limit, indicating that it originates from finite-size effects. By contrast, low-density ferromagnetism remains robust in the square-lattice Hubbard model with hopping along both diagonal directions. We show that this qualitative difference cannot be explained by the degeneracy of the band minima alone, which occurs in both models. Instead, it is controlled by the singular behavior of the density of states at the band bottom: a weak divergence is insufficient to stabilize ferromagnetism, whereas a much stronger quasi-one-dimensional singularity supports a fully polarized ground state even in the dilute limit. Our results demonstrate that band-minimum degeneracy alone is not sufficient for low-density ferromagnetism and that the nature of the band-bottom density-of-states singularity ultimately controls its stability.