In-plane orbital magnetization as a probe for symmetry breaking in strained twisted bilayer graphene

Three symmetries prevent a twisted bilayer of graphene from developing an in-plane spontaneous magnetization in the absence of a magnetic field: time-reversal symmetry, ${C}₂$ symmetry to $\ensuremath{\pi}$ rotation, and ${C}₃$ symmetry to $2\ensuremath{\pi}/3$ rotation. In contrast, there are experimental and theoretical indications that, at certain electron densities, time-reversal and ${C}₂$ symmetries are broken spontaneously, while the ${C}₃$ symmetry is often broken due to strain. We show that in-plane orbital magnetization is a very sensitive probe to the simultaneous breaking of these three symmetries, exhibiting surprisingly large values (of the order of one Bohr magneton per moir\'e unit cell) for valley polarized states at rather small values of heterostrain. We attribute these large values to the large magnitude of the characteristic magnetization of individual Bloch states, which we find to reflect the fast Dirac velocity of single-layer graphene, rather than the slow velocity of the twisted bilayer. These large values are forced to mutually cancel in valley-symmetric states, but the cancellation does not occur in valley-polarized states. Our analysis is carried out both for noninteracting electrons and within a simplified Hartree-Fock framework.