# Examples of anti-symmetric rank-3/4 tensors?

The anti-symmetric rank-3 and rank-4 tensors hold the following properties: \begin{align} \sigma_{mnk} &=-\sigma_{nmk} \\ \sigma_{mnkp} &=-\sigma_{nmkp} \end{align} Note I'm using antisymmetric to mean at least antisymmetric in the first two indices, rather than necessarily having all indices be antisymmetric.

Are there examples of physical properties that correspond to antisymmetric rank-3 and rank-4 tensors? And what about the symmetric rank-3 and rank-4 physical tensors?

• Am I reading to much into the word "the"? Is there a specific anti/symmetric tensor you are looking to calculate? Or are you asking in general how any antisymmetric/symmetric tensor would be calculated? Depending on which is the case, I think the question needs more detail or is a bit too broad
– Tyberius
Apr 12, 2022 at 1:54
• @Tyberius I don't know which physical tensor (in particular, rank-3 and rank-4) will hold the antisymmetric properties. That's what I am looking for. For rank-3 symmetric tensor, the piezomagnetic tensor is an example.
– Jack
Apr 12, 2022 at 2:33
• Oh so you aren't necessarily looking to compute these, but to find examples of physical properties that are anti/symmetric 3rd/4th rank tensors?
– Tyberius
Apr 12, 2022 at 4:11
• Do you mean tensors that are antisymmetric w.r.t. the permutation of the first two indices, or antisymmetric w.r.t. the permutation of any two indices? Apr 12, 2022 at 10:54
• @wzkchem5 Just the first two indices.
– Jack
Apr 12, 2022 at 11:50

In this paper, the authors discuss the following nonlinear current response under an electric field: $$j_a=\chi_{abc}^{int}E_bE_c \tag{1}$$ in which $$\chi_{abc}^{int}=\int \dfrac{d\vec{k}}{(2\pi)^d} \Lambda_{abc}(\vec{k}) \tag{2}$$ with $$\Lambda_{abc}=-\sum_n (\partial_a G_{bc}^n-\partial_b G_{ac}^n)f_0 \tag{3},$$ from which we see $$\chi_{abc}^{int}$$ is a rank-3 anti-symmetric tensor.
In another paper, the authors generalize the response relation $$(1)$$ to third-order (also driven by Berry curvature) and therefore a rank-4 anti-symmetric tensor is found.