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 moiré superlattice's spacing $l_{\text{TF}}\ll l_{\text{moiré}}$. 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-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 $\pm 1/4$. 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) 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.

Figure 1

Lowest one-magnon band spectrum of the TBLG in the chiral limit. Energies are measured with respect to the fully polarized state. The gapless blue curve corresponds to the single-spin flip branch associated with the $\text{SU}\left(2\right)$ breaking Goldstone mode. The gapped red curve corresponds to the single-valley flip branch associated with breaking the discrete time-reversal symmetry. Energies are in units of $\frac{8\pi v_{0}sin(\theta /2)}{3a_0}\approx 0.19\text{eV}$. $k_x$ and $k_y$ are in units of $\frac{8\pi sin(\theta /2)}{3a_0}$. We have used the interaction form $V$ [Eq. (5)].

Figure 2

Lowest one-magnon (single spin-flip) band spectrum of the TBLG as the realistic system is approached. Energies are measured with respect to the fully polarized state in the chiral limit. Energies are in units of $\frac{8\pi v_{0}sin(\theta /2)}{3a_0}\approx 0.19\text{eV}$. Here ${\mathrm{\Delta }}_t={\mathrm{\Delta }}_b=0.1\approx 18\text{meV}$. $k_x$ and $k_y$ are in units of $\frac{8\pi sin(\theta /2)}{3a_0}$.We have used the interaction form $V$ [Eq. (5)].

Figure 3

Spin stiffness associated with the one-magnon band spectrum of the TBLG. A negative spin stiffness signals the instability of the ferromagnetic state. We have used a fit of the form $E={\rho }_s{\left|k\right|}^2$. We have used the interaction form $V$ [Eq. (5)].