Low density ferromagnetism in the Hubbard model

TL;DR

This paper investigates ferromagnetism in the Hubbard model across different dimensions, finding conditions for its emergence in 1D and proving its absence in 3D for certain band structures.

Contribution

It extends the understanding of ferromagnetism in the Hubbard model by analyzing the effects of band structure and dimensionality using analytical and numerical methods.

Findings

Saturated ferromagnetism occurs in 1D above a critical U for specific band structures.

In 3D, ferromagnetism does not occur for non-pathological band structures.

The study confirms and extends previous theoretical scenarios for ferromagnetism in low-dimensional systems.

Abstract

A single-band Hubbard model with nearest and next-nearest neighbour hopping is studied for $d=1$, 2, 3, using both analytical and numerical techniques. In one dimension, saturated ferromagnetism is found above a critical value of $U$ for a band structure with two minima and for small and intermediate densities. This is an extension of a scenario recently proposed by M\"uller--Hartmann. For three dimensions and non-pathological band structures, it is proven that such a scenario does not work.