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.

For identical input frequencies, intrinsic permutation symmetry of nonlinear susceptibility makes the last two Cartesian indices symmetric. Store their six independent pairs as one index, so the quadratic response uses a three-by-six array $d_{iJ}$. In the convention $d_{ijk}={\chi }_{ijk}^{(2)}/2$, the contracted input vector is $(A_x^2,A_y^2,A_z^2,2A_y{A}_z,2A_z{A}_x,2A_x{A}_y{)}^T$. The factors of two preserve the sum over both off-diagonal orderings. This bookkeeping does not assert the stronger Kleinman symmetry and does not change the tensor's physical order.

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.