# Stability of Ferromagnetism in Hubbard Models with Nearly-Flat Bands

## TL;DR

This paper rigorously demonstrates that ferromagnetism in Hubbard models with nearly-flat bands remains stable against single-spin flips when Coulomb interaction is sufficiently large, providing insights into nonperturbative phenomena in electron systems.

## Contribution

It extends the understanding of ferromagnetism stability in Hubbard models by analyzing nearly-flat bands with perturbations, establishing conditions for stability and spin-wave excitations.

## Key findings

- Ferromagnetic state is stable against single-spin flip for large Coulomb U.
- Bounds on dispersion relation confirm healthy spin-wave excitations.
- Results apply to non-singular Hubbard models with nearly-flat bands.

## Abstract

Whether spin-independent Coulomb interaction in an electron system can be the origin of ferromagnetism has been an open problem for a long time. Recently, a "constructive" approach to this problem has been developed, and the existence of ferromagnetism in the ground states of certain Hubbard models was established rigorously. A special feature of these Hubbard models is that their lowest bands (in the corresponding single-electron problems) are completely flat. Here we study models obtained by adding small but arbitrary translation-invariant perturbation to the hopping Hamiltonian of these flat-band models. The resulting models have nearly-flat lowest bands. We prove that the ferromagnetic state is stable against a single-spin flip provided that Coulomb interaction U is sufficiently large. (It is easily found that the same state is unstable against a single-spin flip if U is small enough.) We also prove upper and lower bounds for the dispersion relation of the lowest energy eigenstate with a single flipped spin, which bounds establish that the model has "healthy" spin-wave excitation. It is notable that the (local) stability of ferromagnetism is proved in non-singular Hubbard models, in which we must overcome competition between the kinetic energy and the Coulomb interaction. We also note that this is one of the very few rigorous and robust results which deal with truly nonperturbative phenomena in many electron systems. The local stability strongly suggests that the Hubbard models with nearly flat bands have ferromagnetic ground states. We believe that the present models can be studied as paradigm models for (insulating) ferromagnetism in itinerant electron systems.