# Gauge covariances and nonlinear optical responses

## TL;DR

This paper develops a covariant formalism for calculating nonlinear optical responses in crystals using the reduced density matrix approach, comparing length and velocity gauges, and analyzing the effects of band truncation.

## Contribution

It introduces a covariant derivative representation of the position operator, providing compact expressions for nonlinear responses in both gauges and analyzing gauge equivalence under band truncation.

## Key findings

- Derived compact expressions for first and third order nonlinear responses in graphene.
- Showed that gauge equivalence is broken by band truncation and how to properly implement truncation.
- Compared length and velocity gauge formalisms, highlighting their formal similarities and differences.

## 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.