Higher-order bulk photovoltaic effects, quantum geometry and application to $p$-wave magnets

The injection and shift currents are generalized to the $\ell$th-order injection and shift currents for the longitudinal conductivities in the two-band model, where $\ell$ is the power of the applied electric field. In addition, the formulas for the higher-order injection current are expressed in terms of the quantum metric and the higher-order shift current in terms of the higher-order quantum connection. Then, they are applied to $p$-wave magnets. It is shown that the injection and shift currents are zero. On the other hand, the $\ell$th-order injection and shift currents with odd $\ell$ are nonzero when the direction of the N\'{e}el vector of the $p$-wave magnet points to an in-plane direction.