SHE-c

SHE tensors under Laue groups of category (c)

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Laue group[magnetic point group]	Parity	$s_{x}$	$s_{y}$	$s_{z}$
$2’/m’(2')[2',$ $m',2'/m']$	T-even	${\begin{pmatrix}0 &0&\sigma^{x}_{xz}\\ 0 &0& \sigma^{x}_{yz}\\ \sigma^{x}_{zx} & \sigma^{x}_{zy} & 0\end{pmatrix}}$	${\begin{pmatrix}0 &0&\sigma^{y}_{xz}\\ 0 &0& \sigma^{y}_{yz}\\ \sigma^{y}_{zx} & \sigma^{y}_{zy} & 0\end{pmatrix}}$	${\begin{pmatrix}\sigma^{z}_{xx} & \sigma^{z}_{xy} & 0\\ \sigma^{z}_{yx} & \sigma^{z}_{yy} &0\\ 0 & 0 & \sigma^{z}_{zz}\end{pmatrix}}$
	T-odd	${\begin{pmatrix}\sigma^{x}_{xx} & \sigma^{x}_{xy} & 0\\ \sigma^{x}_{yx} & \sigma^{x}_{yy} &0\\ 0 & 0 & \sigma^{x}_{zz}\end{pmatrix}}$	${\begin{pmatrix}\sigma^{y}_{xx} & \sigma^{y}_{xy} & 0\\ \sigma^{y}_{yx} & \sigma^{y}_{yy} &0\\ 0 & 0 & \sigma^{y}_{zz}\end{pmatrix}}$	${\begin{pmatrix}0 &0&\sigma^{z}_{xz}\\ 0 &0& \sigma^{z}_{yz}\\ \sigma^{z}_{zx} & \sigma^{z}_{zy} & 0\end{pmatrix}}$
$m’m’m(2'2'2)[2'2'2,$ $m'm2',m'm'2,m'm'm]$	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	${\begin{pmatrix}0 &0&\sigma^{x}_{xz}\\ 0 &0& 0\\ \sigma^{x}_{zx} & 0 & 0\end{pmatrix}}$	$\begin{pmatrix}0&0&0\\0&0&\sigma^y_{yz}\\0&\sigma^y_{zy}&0\\\end{pmatrix}$	${\begin{pmatrix}\sigma^{z}_{xx} &0 & 0\\ 0 & \sigma^{z}_{yy} &0\\ 0 & 0 & \sigma^{z}_{zz}\end{pmatrix}}$
$4’/m(4')$ $[4',$ $\overline{4}$ $',4'/m]$	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	${\begin{pmatrix}0 &0&\sigma^{x}_{xz}\\ 0 &0& \sigma^{x}_{yz}\\ \sigma^{x}_{zx} & \sigma^{x}_{zy} & 0\end{pmatrix}}$	${\begin{pmatrix}0 &0&\sigma^{x}_{yz}\\ 0 &0& -\sigma^{x}_{xz}\\ \sigma^{x}_{zy} & -\sigma^{x}_{zx} & 0\end{pmatrix}}$	${\begin{pmatrix}\sigma^{z}_{xx} & \sigma^{z}_{xy} & 0\\ \sigma^{z}_{xy} & -\sigma^{z}_{xx} &0\\ 0 & 0 & 0\end{pmatrix}}$
$4’/mm’m(4'22')$ $[4'2'2,4'm'm,$ $\overline{4}$ $'2'm,$ $\overline{4}$ $'m'2,4'/mm'm]$	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	$\begin{pmatrix}0&0&0\\0&0&\sigma^x_{yz}\\0&\sigma^x_{zy}&0\\\end{pmatrix}$	${\begin{pmatrix}0 &0&\sigma^{x}_{yz}\\ 0 &0& 0\\ \sigma^{x}_{zy} & 0 & 0\end{pmatrix}}$	${\begin{pmatrix}0& \sigma^{z}_{xy} & 0\\\sigma^{z}_{xy} & 0&0\\ 0 & 0 & 0\end{pmatrix}}$
$4/mm’m’(42'2')$ $[42'2',4m'm',$ $\overline{4}$ $2'm',$ $\overline{4}$ $m'2',4/mm'm']$	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	${\begin{pmatrix}0 &0&\sigma^{x}_{xz}\\ 0 &0& 0\\ \sigma^{x}_{zx} & 0 & 0\end{pmatrix}}$	$\begin{pmatrix}0&0&0\\0&0&\sigma^x_{xz}\\0&\sigma^x_{zx}&0\\\end{pmatrix}$	${\begin{pmatrix}\sigma^{z}_{xx} &0 & 0\\ 0 & \sigma^{z}_{xx} &0\\ 0 & 0 & \sigma^{z}_{zz}\end{pmatrix}}$
$\overline{3}$ $1m’(32')[312',31m',$ $\overline{3}$ $1m']$	T-even	${\begin{pmatrix}0 & \sigma^{x}_{xy} & 0\\ \sigma^{x}_{xy} & 0 & \sigma^{x}_{yz}\\ 0 & \sigma^{x}_{zy} & 0\end{pmatrix}}$	${\begin{pmatrix}\sigma^{x}_{xy} & 0 & -\sigma^{x}_{yz}\\ 0& -\sigma^{x}_{xy} &0\\-\sigma^{x}_{zy} & 0& 0\end{pmatrix}}$	${\begin{pmatrix}0& \sigma^{z}_{xy} & 0\\-\sigma^{z}_{xy} & 0&0\\ 0 & 0 & 0\end{pmatrix}}$
	T-odd	${\begin{pmatrix}\sigma^{x}_{xx} & 0 & \sigma^{x}_{xz}\\ 0& -\sigma^{x}_{xx} &0\\\sigma^{x}_{zx} & 0& 0\end{pmatrix}}$	${\begin{pmatrix}0 & -\sigma^{x}_{xx} & 0\\ -\sigma^{x}_{xx} & 0 & \sigma^{x}_{xz}\\ 0 & \sigma^{x}_{zx} & 0\end{pmatrix}}$	${\begin{pmatrix}\sigma^{z}_{xx} &0 & 0\\ 0 & \sigma^{z}_{xx} &0\\ 0 & 0 & \sigma^{z}_{zz}\end{pmatrix}}$
$6’/m’(6')[6',$ $\overline{6}$ $',6'/m']$	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	${\begin{pmatrix}\sigma^{x}_{xx} & \sigma^{x}_{xy} & 0\\ \sigma^{x}_{xy} & -\sigma^{x}_{xx} &0\\ 0 & 0 & 0\end{pmatrix}}$	${\begin{pmatrix}\sigma^{x}_{xy} & -\sigma^{x}_{xx} & 0\\ -\sigma^{x}_{xx} & -\sigma^{x}_{xy} &0\\ 0 & 0 & 0\end{pmatrix}}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$
$6’/m’m’m(6'22')[6'2'2,$ $6'm'm,$ $\overline{6}$ $'m'2,$ $\overline{6}$ $'2'm,6'/m'm'm]$	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	${\begin{pmatrix}0& \sigma^{x}_{xy} & 0\\\sigma^{x}_{xy} & 0&0\\ 0 & 0 & 0\end{pmatrix}}$	${\begin{pmatrix}\sigma^{x}_{xy}& 0 & 0\\0&-\sigma^{x}_{xy}&0\\ 0 & 0 & 0\end{pmatrix}}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$
$6/mm’m’(62'2')[62'2',$ $6m'm',$ $\overline{6}$ $m'2',$ $\overline{6}$ $2'm',6/mm'm']$	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	${\begin{pmatrix}0 &0&\sigma^{x}_{xz}\\ 0 &0& 0\\ \sigma^{x}_{zx} & 0 & 0\end{pmatrix}}$	$\begin{pmatrix}0&0&0\\0&0&\sigma^x_{xz}\\0&\sigma^x_{zx}&0\\\end{pmatrix}$	${\begin{pmatrix}\sigma^{z}_{xx} &0 & 0\\ 0 & \sigma^{z}_{xx} &0\\ 0 & 0 & \sigma^{z}_{zz}\end{pmatrix}}$
$m$ $\overline{3}$ $m’(4'32')[4'32',$ $\overline{4}$ $'3m',m$ $\overline{3}$ $m']$	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.