Is there a bound on the nonlinear optical response of crystals?

The second-order nonlinear response, defined by the relation, J^((2)) (ω±ω)=σ^((2) ) (ω)E_ω^2, is allowed only in media without a center of inversion. A new class of inversion breaking materials, “Weyl semimetals,” have been synthesized and are currently under intense investigation. Recently we reported that the first of this class of materials to be discovered, the transition metal monopnictides such as TaAs, and NbAs, exhibit the largest σ^((2) ) (ω)E_ω^2 of any known crystal. Large values of σ^((2) ) (ω) are of interest for applications involving frequency generation and conversion of light to electrical current. The observation of a “giant” response raises two related questions for which I will attempt to provide partial answers: what is special about TaAs, and is there an upper bound on σ^((2) ) (ω) of inversion-breaking crystal?