Linear and nonlinear optical responses in Green's function formula

Linear and nonlinear optical effect has been widely discussed in large quantity of materials using theoretical or experimental methods. Except linear optical conductivity, higher-order nonlinear responses are not studied fully. Starting from density operator method, we derive optical conductivities of different orders in Green's function formula, and also connect them to novel physical quantities, such as Berry curvature, Berry curvature dipole, third-order nonlinear Hall conductivity and so on. Based on the advantages of Green's function formulas, we believe that these formulas have a lot of benefits for many-body effect study in high-order nonlinear optical responses.