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

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.