# Ferromagnetism in narrow bands of moir\'e superlattices

## TL;DR

This paper demonstrates that nearly flat narrow bands in various moiré superlattices can host robust ferromagnetic insulators at integer fillings, with stability influenced by band flatness, interactions, and quantum geometry.

## Contribution

It provides analytical and numerical evidence for ferromagnetism in flat bands of moiré materials, extending results to finite bandwidth and larger systems using DMRG and exact diagonalization.

## Key findings

- Maximally spin and valley polarized insulators at integer fillings in flat bands.
- Interactions can destabilize ferromagnetic states by inducing effective dispersive terms.
- Results applicable even when valley Chern number is zero, emphasizing the role of quantum geometry.

## Abstract

Many graphene moir\'e superlattices host narrow bands with non-zero valley Chern numbers. We provide analytical and numerical evidence for a robust spin and/or valley polarized insulator at total integer band filling in nearly flat bands of several different moir\'e materials. In the limit of a perfectly flat band, we present analytical arguments in favor of the ferromagnetic state substantiated by numerical calculations. Further, we numerically evaluate its stability for a finite bandwidth. We provide exact diagonalization results for models appropriate for ABC trilayer graphene aligned with hBN, twisted double bilayer graphene, and twisted bilayer graphene aligned with hBN. We also provide DMRG results for a honeycomb lattice with a quasi-flat band and non-zero Chern number, which extend our results to larger system sizes. We find a maximally spin and valley polarized insulator at all integer fillings when the band is sufficiently flat. We also show that interactions may induce effective dispersive terms strong enough to destabilize this state. These results still hold in the case of zero valley Chern number (for example, trivial side of TLG/hBN). We give an intuitive picture based on extended Wannier orbitals, and emphasize the role of the quantum geometry of the band, whose microscopic details may enhance or weaken ferromagnetism in moir\'e materials.