  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.
