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

We study ferromagnetism and its stability in twisted bilayer graphene. We show that in the perfectly flat band limit and at filling fractions $\pm 3/4$, the saturated ferromagnetic (spin and valley polarized) states are ideal ground state 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 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 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 value of spin-stiffness can determine the energy of charged skyrmions. We further calculate the spectrum of the gapped single valley-flip excitations. The valley-mode gap can be used to estimate the transition temperature of quantum anomalous Hall state.