Gauge covariances and nonlinear optical responses

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 requires a revision of the equations of motion of the reduced density matrix in the velocity gauge.

Figure 1

A conceptual picture of an effective Hamiltonian that describes the bands in subspace ${\mathcal{E}}_0$. The Fermi level lies somewhere in that subspace. For bands inside ${\mathcal{E}}_0$, the energy difference is of order $\delta$, while the energy difference between bands in different subspaces is of order $\mathrm{\Delta }\gg \delta$.