Theoretical calculations of nonlinear optical calculations of 2D materials

One important feature of two dimensional (2D) materials is that they possess an exceptional nonlinear optical (NLO) response to light, with conduc¬tivities that are several orders of magnitude larger than their 3D counterparts. The theoretical descriptions of these NLO responses in crystalline systems in¬volve two different representations of the perturbation: the length and velocity gauges. The former has been the formalism of choice for the past two decades; the latter was implemented only recently, due to concerns that it could not be pratically implemented without breaking sum rules – a set of identities that en¬sure the equivalence between the two formalisms – which would then render the results unphysical. In this work, we shall review and summarize our contri¬butions to the study of the two formalisms and of their relationship by means of the aforementioned sum rules.