Optical response and nonlinear Hall response of twisted bilayer graphene in the insulating state

In this work, we calculate the optical response and the nonlinear Hall response of twisted bilayer graphene (TBG) in the insulating states. Different insulating states, including spin-valley polarized (SPVP) state, spin-polarized quantum Hall(SPQH) state, spin-polarized valley Hall (SPVH) state, and spin polarized Kramers-intervalley coherence (SPKIVC) state, are considered. We calculate the optical conductivity ($\sigma_{xx}$) of these four states and we find that the spin-valley polarized state shows a two-peak feature which is different from the other three states. In order to distinguish the other three states, we calculate the nonlinear Hall response which is proportional to Berry curvature dipole, and we find the chemical potential dependence of the Berry curvature dipole can distinguish these three states. Our result can be tested in the spectroscopy and transport experiments and may be helpful to determine the nature of insulating state in TBG.