What are the material restrictions for a $chi^{3}$ non-linear optical material?

---

Long answer:

Basically, $chi^{(3)}$ is a rank four tensor with 81 components. You then apply the Neuman principle saying that symmetries of the crystal that supports $chi^{(3)}$ should also be the symmetries of the tensor. Then you look for independent components of this tensor, which boils down to looking for trivial irreducible sub-representations in the representation of the symmetry group of the crystal over that rank four tensor.

So lets say the representation of your crystal symmetry group ($G$) over vectors is given by: $Mleft(aright): mathbb{R}^3to mathbb{R}^3$, where $ain G$

Then the induced representation for $chi^{(3)}$ is $Kleft(aright)=Mleft(aright)otimes Mleft(aright) otimes Mleft(aright) otimes Mleft(aright)$. If you want to know the independent components of $chi^{(3)}$, simply define the projection operator, to project into trivial irreps:

$P=frac{1}{#G}sum_{ain G} Kleft(aright)$

$P:mathbb{R}^3 otimes mathbb{R}^3 otimes mathbb{R}^3 otimes mathbb{R}^3tomathbb{R}^3 otimes mathbb{R}^3 otimes mathbb{R}^3 otimes mathbb{R}^3 $

The eigenvectors, with eigenvalues=1, of this projection operator will be the allowed components of $chi^{(3)}$, if you only want to know the number of allowed components $n_{allowed}$, it is given by the trace of the projection operator:

$n_{allowed}=Trleft(Pright)=frac{1}{#G}sum_{ain G} Trleft(Kleft(aright)right)=frac{1}{#G}sum_{ain G} Trleft(Mleft(aright)right)^4$

---

Short answer:

Have a look in a suitable nonlinear optics book, e.g. Popov, Svirko, Zheludev "Susceptibility Tensors for Nonlinear Optics", Appendix F. They list all the allowed components of $chi^{(3)}$ for all crystallographic classes. There is no class where $chi^{(3)}$ is not allowed, but in some cases the number of allowed components can be as low as 2 (Cubic system, class 432) or even 1 (isotropic medium).