Nonlinear-susceptibility tensor symmetry

The order-$n$ response has one output vector index and $n$ input vector indices. Material point-group invariance, permutation of input index-frequency pairs, and reality of the electric polarization impose different constraints. They must not be conflated with the stronger dispersion-neglecting Kleinman symmetry.

An isotropic fourth-rank polar tensor has the displayed form. If its three input slots are fully interchangeable at the same relevant frequency arguments, then $a=b=c$, giving ${\chi }_{xxxx}=3{\chi }_{xxyy}$. For mixed-sign or unequal frequencies, intrinsic permutation exchanges the arguments too; it need not make all coefficients at one fixed ordering equal.

Far from relevant resonances, when frequency dispersion of the nonlinear response can be neglected, a susceptibility can acquire full permutation symmetry among spatial indices. For second order this reduces a general symmetric-input tensor's 18 components to 10 before material point-group reductions. Mere transparency does not establish this approximation. The name refers to D. A. Kleinman.^[ref]

Commuting input field factors permit the response kernel to be symmetrized under interchange of input index-frequency pairs. For example ${\chi }_{ijk}^{(2)}({\omega }_s;{\omega }_1,{\omega }_2)={\chi }_{ikj}^{(2)}({\omega }_s;{\omega }_2,{\omega }_1)$. This does not authorize exchanging indices while keeping unequal frequency arguments in their old slots, nor exchanging the output index without further assumptions.