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

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