Inversion Symmetry in Second Harmonic Generation (SHG)

Question

I have been reading a paper which uses Second Harmonic Generation (SHG) to probe magnetic order in a magnetic material and as I come from more of a condensed matter background, I was hung up on symmetry arguments which seem to be taken as evident.

Specifically, I have repeatedly seen the argument that, for centrosymmetric media, and the electric dipole contribution, the bulk SHG must vanish since under inversion $P\rightarrow-P$ and $E\rightarrow -E$, meaning under inversion a term like $P=\chi^{(2)}E^2$ necessitates $\chi^{(2)}=0$. What part of this argument is specific to centrosymmetric media? Why is this not applicable to systems which break inversion symmetry?

Answer 1

It is not exactly clear to me what the issue is, but let me try to clarify. A centrosymmetric medium is symmetric under the inversion of the spatial directions. For that reason, we have $\chi^{(2)}=0$, because it is a tensor with an odd number of indices, which would lead to a minus sign when one inverts the three directions. As a consequence, we can also have that the polarization and electric fields pick up minus sign when one inverts the directions in space, but that is not directly related to the properties of the medium.