Title: Gauge covariances and nonlinear optical responses,
Abstract: The formalism of the reduced density matrix is pursued in both length and
velocity gauges of the perturbation to the crystal Hamiltonian. The covariant
derivative is introduced as a convenient representation of the position
operator. This allow us to write compact expressions for the reduced density
matrix in any order of the perturbation which simplifies the calculations of
nonlinear optical responses; as an example, we compute the first and third
order contributions of the monolayer graphene. Expressions obtained in both
gauges share the same formal structure, allowing a comparison of the effects of
truncation to a finite set of bands. This truncation breaks the equivalence
between the two approaches: its proper implementation can be done directly in
the expressions derived in the length gauge, but require a revision of the
equations of motion of the reduced density matrix in the velocity gauge.