Ferromagnetism and its stability from the one-magnon spectrum in twisted bilayer graphene
Yahya Alavirad, Jay D. Sau
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 ${l}_{g}\ensuremath{\ll}{l}_{\text{TF}}$ and much smaller than the moir\'e superlattice's spacing ${l}_{\text{TF}}\ensuremath{\ll}{l}_{\text{moir\'e}}$. We show that in the perfectly flat band ``chiral'' limit and at filling fractions $\ifmmode\pm\else\textpm\fi{}3/4$, 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 sublattice potential, the same argument can be applied to filling fractions $\ifmmode\pm\else\textpm\fi{}1/4$. 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) toward 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.