Contribution of remote bands to orbital magnetization in twisted bilayer graphene

Motivated by recent theoretical and experimental works on orbital magnetization $M_{\mathrm{orb}}$ for the interacting system, we develop a gauge-invariant framework to compute $M_{\mathrm{orb}}$ for correlated phases of magic-angle twisted bilayer graphene within self-consistent Hartree-Fock approximation. Based on the projector formulation of the theory of orbital magnetization, we evaluate both $M_{\mathrm{orb}}$ and the self-rotation contribution $m_{\mathrm{SR}}$ directly from the Hartree-Fock Hamiltonian. We demonstrate that, in contrast to topological invariants such as the Chern number, both $M_{\mathrm{orb}}$ and $m_{\mathrm{SR}}$ obtain substantial contributions from remote bands and thus require careful convergence with respect to the number of included remote bands. Applying this approach to correlated phases at integer fillings, we obtain converged $M_{\mathrm{orb}}$ and $m_{\mathrm{SR}}$ for time reversal symmetry broken Chern insulating states at $\nu=\pm3$ and for competing correlated phases at other integer fillings. Our results establish a systematic and controlled approach for evaluating orbital magnetization in correlated moir\'e systems and clarify the crucial role of remote bands in determining their magnetic response.