Estimating the Hubbard repulsion sufficient for the onset of nearly-flat-band ferromagnetism

TL;DR

This paper estimates the critical Hubbard repulsion strength needed to induce ferromagnetism in nearly-flat-band models, extending a rigorous method and comparing it with finite-size scaling results under different boundary conditions.

Contribution

The authors extend Tasaki's rigorous method to estimate the ferromagnetic threshold in nearly-flat-band Hubbard models and compare it with finite-size scaling results, highlighting the usefulness of open and periodic boundary conditions.

Findings

Finite-size scaling results align with the extended rigorous estimates.

Open and periodic boundary conditions are particularly effective for analysis.

The threshold $U_{th}$ can be estimated reliably using the combined approach.

Abstract

We consider nearly-flat-band Hubbard models of a ferromagnet, that is the models that are weak perturbations of those flat-band Hubbard models whose ground state is ferromagnetic for any nonzero strength $U$ of the Hubbard repulsion. In contrast to the flat-band case, in the nearly-flat-band case the ground state, being paramagnetic for $U$ in a vicinity of zero, turns into a ferromagnetic one only if $U$ exceeds some nonzero threshold value $U_{th}$. We address the question whether $U_{th}$ of the considered models is in a physical range, therefore we attempt at obtaining possibly good estimates of the threshold value $U_{th}$. A rigorous method proposed by Tasaki is extended and the resulting estimates are compared with small-system, finite-size scaling results obtained for open- and periodic-boundary conditions. Contrary to suggestions in literature, we find the latter conditions particularly useful for our task.