SHE-b

SHE tensor under Laue groups of category (b)

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Laue group [magnetic point group]	Parity	$s_{x}$	$s_{y}$	$s_{z}$
$\overline{1}$ $(1)[1,$ $\overline{1}$ $]$	T-even	${\begin{pmatrix}\sigma^{x}_{xx} & \sigma^{x}_{xy} & \sigma^{x}_{xz}\\ \sigma^{x}_{yx} & \sigma^{x}_{yy} & \sigma^{x}_{yz}\\ \sigma^{x}_{zx} & \sigma^{x}_{zy} & \sigma^{x}_{zz}\end{pmatrix}}$	$\begin{pmatrix}\sigma^{y}_{xx}&\sigma^{y}_{xy}&\sigma^{y}_{xz}\\ \sigma^{y}_{yx}&\sigma^{y}_{yy}&\sigma^{y}_{yz}\\ \sigma^{y}_{zx}&\sigma^{y}_{zy}&\sigma^{y}_{zz}\end{pmatrix}$	$\begin{pmatrix}\sigma^{z}_{xx}&\sigma^{z}_{xy}&\sigma^{z}_{xz}\\ \sigma^{z}_{yx}&\sigma^{z}_{yy}&\sigma^{z}_{yz}\\ \sigma^{z}_{zx}&\sigma^{z}_{zy}&\sigma^{z}_{zz}\end{pmatrix}$
	T-odd	${\begin{pmatrix}\sigma^{x}_{xx} & \sigma^{x}_{xy} & \sigma^{x}_{xz}\\ \sigma^{x}_{yx} & \sigma^{x}_{yy} & \sigma^{x}_{yz}\\ \sigma^{x}_{zx} & \sigma^{x}_{zy} & \sigma^{x}_{zz}\end{pmatrix}}$	$\begin{pmatrix}\sigma^{y}_{xx}&\sigma^{y}_{xy}&\sigma^{y}_{xz}\\ \sigma^{y}_{yx}&\sigma^{y}_{yy}&\sigma^{y}_{yz}\\ \sigma^{y}_{zx}&\sigma^{y}_{zy}&\sigma^{y}_{zz}\end{pmatrix}$	$\begin{pmatrix}\sigma^{z}_{xx}&\sigma^{z}_{xy}&\sigma^{z}_{xz}\\ \sigma^{z}_{yx}&\sigma^{z}_{yy}&\sigma^{z}_{yz}\\ \sigma^{z}_{zx}&\sigma^{z}_{zy}&\sigma^{z}_{zz}\end{pmatrix}$
$2/m(2)[2,m,2/m]$	T-even	${\begin{pmatrix}0 &0&\sigma^{x}_{xz}\\ 0 &0& \sigma^{x}_{yz}\\ \sigma^{x}_{zx} & \sigma^{x}_{zy} & 0\end{pmatrix}}$	${\begin{pmatrix}0 &0&\sigma^{y}_{xz}\\ 0 &0& \sigma^{y}_{yz}\\ \sigma^{y}_{zx} & \sigma^{y}_{zy} & 0\end{pmatrix}}$	${\begin{pmatrix}0 &0&\sigma^{y}_{xz}\\ 0 &0& \sigma^{y}_{yz}\\ \sigma^{y}_{zx} & \sigma^{y}_{zy} & 0\end{pmatrix}}$
	T-odd	${\begin{pmatrix}0 &0&\sigma^{x}_{xz}\\ 0 &0& \sigma^{x}_{yz}\\ \sigma^{x}_{zx} & \sigma^{x}_{zy} & 0\end{pmatrix}}$	${\begin{pmatrix}0 &0&\sigma^{y}_{xz}\\ 0 &0& \sigma^{y}_{yz}\\ \sigma^{y}_{zx} & \sigma^{y}_{zy} & 0\end{pmatrix}}$	${\begin{pmatrix}0 &0&\sigma^{y}_{xz}\\ 0 &0& \sigma^{y}_{yz}\\ \sigma^{y}_{zx} & \sigma^{y}_{zy} & 0\end{pmatrix}}$
$mmm(222)[222,mm2,mmm]$	T-even	$\begin{pmatrix}0&0&0\\0&0&\sigma^x_{yz}\\0&\sigma^x_{zy}&0\\\end{pmatrix}$	${\begin{pmatrix}0 &0& \sigma^{y}_{xz}\\ 0 &0& 0\\ \sigma^{y}_{zx}&0 & 0\end{pmatrix}}$	$\begin{pmatrix}0&\sigma^z_{xy}&0\\\sigma^z_{yx}&0&0\\0&0&0\\\end{pmatrix}$
	T-odd	$\begin{pmatrix}0&0&0\\0&0&\sigma^x_{yz}\\0&\sigma^x_{zy}&0\\\end{pmatrix}$	$\begin{pmatrix}0&0&\sigma^x_{xz}\\0&0&0\\\sigma^x_{zx}&0&0\\\end{pmatrix}$	${\begin{pmatrix}0& \sigma^{z}_{xy} & 0\\\sigma^{z}_{yx} & 0&0\\ 0 & 0 & 0\end{pmatrix}}$
$4/m(4)[4,$ $\overline{4}$ $,4/m]$	T-even	${\begin{pmatrix}0 &0&\sigma^{x}_{xz}\\ 0 &0& \sigma^{x}_{yz}\\ \sigma^{x}_{zx} & \sigma^{x}_{zy} & 0\end{pmatrix}}$	${\begin{pmatrix}0 &0&-\sigma^{x}_{yz}\\ 0 &0& \sigma^{x}_{xz}\\ -\sigma^{x}_{zy} & \sigma^{x}_{zx} & 0\end{pmatrix}}$	${\begin{pmatrix}\sigma^{z}_{xx} & \sigma^{z}_{xy} & 0\\ -\sigma^{z}_{xy} & \sigma^{z}_{xx} &0\\ 0 & 0 & \sigma^{z}_{zz}\end{pmatrix}}$
	T-odd	${\begin{pmatrix}0 &0&\sigma^{x}_{xz}\\ 0 &0& \sigma^{x}_{yz}\\ \sigma^{x}_{zx} & \sigma^{x}_{zy} & 0\end{pmatrix}}$	${\begin{pmatrix}0 &0&-\sigma^{x}_{yz}\\ 0 &0& \sigma^{x}_{xz}\\ -\sigma^{x}_{zy} & \sigma^{x}_{zx} & 0\end{pmatrix}}$	${\begin{pmatrix}\sigma^{z}_{xx} & \sigma^{z}_{xy} & 0\\ -\sigma^{z}_{xy} & \sigma^{z}_{xx} &0\\ 0 & 0 & \sigma^{z}_{zz}\end{pmatrix}}$
$4/mmm(422)[422,4mm,$ $\overline{4}$ $2m,$ $\overline{4}$ $m2,4/mmm]$	T-even	$\begin{pmatrix}0&0&0\\0&0&\sigma^x_{yz}\\0&\sigma^x_{zy}&0\\\end{pmatrix}$	${\begin{pmatrix}0 &0&-\sigma^{x}_{yz}\\ 0 &0& 0\\ -\sigma^{x}_{zy} & 0 & 0\end{pmatrix}}$	${\begin{pmatrix}0& \sigma^{z}_{xy} & 0\\-\sigma^{z}_{xy} & 0&0\\ 0 & 0 & 0\end{pmatrix}}$
	T-odd	$\begin{pmatrix}0&0&0\\0&0&\sigma^x_{yz}\\0&\sigma^x_{zy}&0\\\end{pmatrix}$	${\begin{pmatrix}0 &0&-\sigma^{x}_{yz}\\ 0 &0& 0\\ -\sigma^{x}_{zy} & 0 & 0\end{pmatrix}}$	${\begin{pmatrix}0& \sigma^{z}_{xy} & 0\\-\sigma^{z}_{xy} & 0&0\\ 0 & 0 & 0\end{pmatrix}}$
$\overline{3}$ $(3)[3,$ $\overline{3}$ $]$	T-even	${\begin{pmatrix}\sigma^{x}_{xx} & \sigma^{y}_{xx} & \sigma^{x}_{xz}\\ \sigma^{y}_{xx} & -\sigma^{x}_{xx} & -\sigma^{y}_{xz}\\ \sigma^{x}_{zx} & -\sigma^{y}_{zx} & 0\end{pmatrix}}$	${\begin{pmatrix}\sigma^{y}_{xx} &- \sigma^{x}_{xx} & \sigma^{y}_{xz}\\ \sigma^{x}_{xx} & -\sigma^{y}_{xx} & \sigma^{x}_{xz}\\ \sigma^{y}_{zx} & \sigma^{x}_{zx} & 0\end{pmatrix}}$	${\begin{pmatrix}\sigma^{z}_{xx} & -\sigma^{z}_{yx} & 0\\ \sigma^{z}_{yx} & \sigma^{z}_{xx} &0\\ 0 & 0 & \sigma^{z}_{zz}\end{pmatrix}}$
	T-odd	${\begin{pmatrix}\sigma^{x}_{xx} & \sigma^{y}_{xx} & \sigma^{x}_{xz}\\ \sigma^{y}_{xx} & -\sigma^{x}_{xx} & -\sigma^{y}_{xz}\\ \sigma^{x}_{zx} & -\sigma^{y}_{zx} & 0\end{pmatrix}}$	${\begin{pmatrix}\sigma^{y}_{xx} &- \sigma^{x}_{xx} & \sigma^{y}_{xz}\\ \sigma^{x}_{xx} & -\sigma^{y}_{xx} & \sigma^{x}_{xz}\\ \sigma^{y}_{zx} & \sigma^{x}_{zx} & 0\end{pmatrix}}$	${\begin{pmatrix}\sigma^{z}_{xx} & -\sigma^{z}_{yx} & 0\\ \sigma^{z}_{yx} & \sigma^{z}_{xx} &0\\ 0 & 0 & \sigma^{z}_{zz}\end{pmatrix}}$
$\overline{3}$ $1m(32)[312,31m,$ $\overline{3}$ $1m]$	T-even	${\begin{pmatrix}\sigma^{x}_{xx} & 0& 0\\0 &- \sigma^{x}_{xx} &\sigma^{x}_{yz}\\ 0 & \sigma^{x}_{zy} & 0\end{pmatrix}}$	${\begin{pmatrix}0 & -\sigma^{x}_{xx} & -\sigma^{x}_{yz}\\ -\sigma^{x}_{xx} & 0&0\\ -\sigma^{x}_{zy} & 0& 0\end{pmatrix}}$	${\begin{pmatrix}0& \sigma^{z}_{xy} & 0\\ -\sigma^{z}_{xy} & 0&0\\ 0 & 0 & 0\end{pmatrix}}$
	T-odd	${\begin{pmatrix}\sigma^{x}_{xx} & 0& 0\\0 &- \sigma^{x}_{xx} &\sigma^{x}_{yz}\\ 0 & \sigma^{x}_{zy} & 0\end{pmatrix}}$	${\begin{pmatrix}0 & -\sigma^{x}_{xx} & -\sigma^{x}_{yz}\\ -\sigma^{x}_{xx} & 0&0\\ -\sigma^{x}_{zy} & 0& 0\end{pmatrix}}$	${\begin{pmatrix}0& \sigma^{z}_{xy} & 0\\ -\sigma^{z}_{xy} & 0&0\\ 0 & 0 & 0\end{pmatrix}}$
$6/m(6)[6,$ $\overline{6}$ $,6/m]$	T-even	${\begin{pmatrix}0 &0&\sigma^{x}_{xz}\\ 0 &0& \sigma^{x}_{yz}\\ \sigma^{x}_{zx} & \sigma^{x}_{zy} & 0\end{pmatrix}}$	${\begin{pmatrix}0 &0&-\sigma^{x}_{yz}\\ 0 &0& \sigma^{x}_{xz}\\ -\sigma^{x}_{zy} & \sigma^{x}_{zx} & 0\end{pmatrix}}$	${\begin{pmatrix}\sigma^{z}_{xx} & \sigma^{z}_{xy} & 0\\ -\sigma^{z}_{xy} & \sigma^{z}_{xx} &0\\ 0 & 0 & \sigma^{z}_{zz}\end{pmatrix}}$
	T-odd	${\begin{pmatrix}0 &0&\sigma^{x}_{xz}\\ 0 &0& \sigma^{x}_{yz}\\ \sigma^{x}_{zx} & \sigma^{x}_{zy} & 0\end{pmatrix}}$	${\begin{pmatrix}0 &0&-\sigma^{x}_{yz}\\ 0 &0& \sigma^{x}_{xz}\\ -\sigma^{x}_{zy} & \sigma^{x}_{zx} & 0\end{pmatrix}}$	${\begin{pmatrix}\sigma^{z}_{xx} & \sigma^{z}_{xy} & 0\\ -\sigma^{z}_{xy} & \sigma^{z}_{xx} &0\\ 0 & 0 & \sigma^{z}_{zz}\end{pmatrix}}$
$6/mmm(622)[622,6mm,$ $\overline{6}$ $m2,$ $\overline{6}$ $2m,6/mmm]$	T-even	$\begin{pmatrix}0&0&0\\0&0&\sigma^x_{yz}\\0&\sigma^x_{zy}&0\\\end{pmatrix}$	${\begin{pmatrix}0 &0&\sigma^{x}_{yz}\\ 0 &0& 0\\ -\sigma^{x}_{zy} & 0 & 0\end{pmatrix}}$	${\begin{pmatrix}0& \sigma^{z}_{xy} & 0\\-\sigma^{z}_{xy} & 0&0\\ 0 & 0 & 0\end{pmatrix}}$
	T-odd	$\begin{pmatrix}0&0&0\\0&0&\sigma^x_{yz}\\0&\sigma^x_{zy}&0\\\end{pmatrix}$	${\begin{pmatrix}0 &0&\sigma^{x}_{yz}\\ 0 &0& 0\\ -\sigma^{x}_{zy} & 0 & 0\end{pmatrix}}$	${\begin{pmatrix}0& \sigma^{z}_{xy} & 0\\-\sigma^{z}_{xy} & 0&0\\ 0 & 0 & 0\end{pmatrix}}$
$m$ $\overline{3}$ $(23)[23,m$ $\overline{3}$ $]$	$T-even	${\begin{pmatrix}0 &0&0\\ 0 &0& \sigma^{y}_{zx}\\ 0& \sigma^{y}_{xz} & 0\end{pmatrix}}$	${\begin{pmatrix}0 &0&\sigma^{y}_{xz}\\ 0 &0& 0\\ \sigma^{y}_{zx}& 0 & 0\end{pmatrix}}$	${\begin{pmatrix}0& \sigma^{y}_{zx} & 0\\ \sigma^{y}_{xz} & 0&0\\ 0 & 0 & 0\end{pmatrix}}$
	T-odd	${\begin{pmatrix}0 &0&0\\ 0 &0& \sigma^{y}_{zx}\\ 0& \sigma^{y}_{xz} & 0\end{pmatrix}}$	${\begin{pmatrix}0 &0&\sigma^{y}_{xz}\\ 0 &0& 0\\ \sigma^{y}_{zx}& 0 & 0\end{pmatrix}}$	${\begin{pmatrix}0& \sigma^{y}_{zx} & 0\\ \sigma^{y}_{xz} & 0&0\\ 0 & 0 & 0\end{pmatrix}}$
$m$ $\overline{3}$ $m(432)$ $[432,$ $\overline{4}$ $3m,m$ $\overline{3}$ $m]$	T-even	${\begin{pmatrix}0 &0&0\\ 0 &0&- \sigma^{y}_{xz}\\ 0& \sigma^{y}_{xz} & 0\end{pmatrix}}$	${\begin{pmatrix}0 &0&\sigma^{y}_{xz}\\ 0 &0& 0\\ -\sigma^{y}_{xz} & 0 & 0\end{pmatrix}}$	${\begin{pmatrix}0& -\sigma^{y}_{xz} & 0\\ \sigma^{y}_{xz} & 0&0\\ 0 & 0 & 0\end{pmatrix}}$
	T-odd	${\begin{pmatrix}0 &0&0\\ 0 &0&- \sigma^{y}_{xz}\\ 0& \sigma^{y}_{xz} & 0\end{pmatrix}}$	${\begin{pmatrix}0 &0&\sigma^{y}_{xz}\\ 0 &0& 0\\ -\sigma^{y}_{xz} & 0 & 0\end{pmatrix}}$	${\begin{pmatrix}0& -\sigma^{y}_{xz} & 0\\ \sigma^{y}_{xz} & 0&0\\ 0 & 0 & 0\end{pmatrix}}$

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References:

Symmetry-imposed shape of linear response tensors | Phys. Rev. B
Efficient Electrical Spin Splitter Based on Nonrelativistic Collinear Antiferromagnetism | Phys. Rev. Lett.