# Ferromagnetism and its stability from the one-magnon spectrum in twisted bilayer graphene

## TL;DR

This paper investigates the stability of ferromagnetism in twisted bilayer graphene by analyzing the one-magnon spectrum, providing insights into the conditions for ferromagnetic ground states and their excitations.

## Contribution

It introduces a method to estimate ferromagnetic stability in twisted bilayer graphene using the one-magnon spectrum and extends results from idealized models to more realistic systems.

## Key findings

- Ferromagnetic states are stable at specific filling fractions in the chiral limit.
- Negative magnon excitation energy indicates instability of ferromagnetic states.
- Calculated spin-stiffness informs about skyrmion excitation energies.

## Abstract

We study ferromagnetism and its stability in twisted bilayer graphene. We work with a Hubbard-like interaction that corresponds to the screened Coulomb interaction in a well-defined limit where the Thomas-Fermi screening length $l_\text{TF}$ is much larger than monolayer graphene's lattice spacing $l_g \ll l_\text{TF}$ and much smaller than the Moir\'e super lattice's spacing $ l_\text{TF} \ll l_{\text{Moir\'e}}$. We show that in the perfectly flat band "chiral" limit and at filling fractions $\pm 3/4$, the saturated ferromagnetic (spin and valley polarized) states are ideal ground states candidates in the large band-gap limit. By assuming a large enough substrate (hBN) induced sub-lattice potential, the same argument can be applied to filling fractions $\pm 1/4$. We estimate the regime of stability of the ferromagnetic phase around the chiral limit by studying the exactly calculated spectrum of one-magnon excitations. The instability of the ferromagnetic state is signaled by a negative magnon excitation energy. This approach allows us to deform the results of the idealized chiral model (by increasing the bandwidth and/or modified interactions) towards more realistic systems. Furthermore, we use the low energy part of the exact one-magnon spectrum to calculate the spin-stiffness of the Goldstone modes throughout the ferromagnetic phase. The calculated value of spin-stiffness can determine the excitation energy of charged skyrmions.