SHE-a

SHE tensors under Laue groups of category (a)

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Laue group [magnetic point group]	T	$s_{x}$	$s_{y}$	$s_{z}$
$\overline{1}$ $1’(1')$ $[$ $\overline{1}$ $1',$ $\overline{1}$ $',11']$	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	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$
$2/m1’(21')$ $[21',m1',2/m1',2'/m,2/m']$	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	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$
$mmm1’(2221')$ $[2221',mm21',m'mm,mmm1',m'm'm']$	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	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$
$4/m1’(41')$ $[41',$ $\overline{4}$ $1',4/m',4/m1',4'/m']$	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	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$
$4/mmm1’(4221')$ $[4221',4mm1',$ $\overline{4}$ $2m1',$ $\overline{4}$ $m21', 4/m'mm, 4'/m'm'm,$ $4/mmm1',4'/m'mm',4/m'm'm']$	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	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$
$\overline{3}$ $1’(3')$ $[31',$ $\overline{3}$ $',$ $\overline{3}$ $1']$	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	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$
$\overline{3}$ $1m1’(3'2)$ $[3121',31m1',$ $\overline{3}$ $'1m,$ $\overline{3}$ $'1m',$ $\overline{3}$ $1m1']$	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	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$
$6/m1’(61')[61',$ $\overline{6}$ $1',6'/m,6/m',6/m1']$	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	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$
$6/mmm1’(6221')[6221',6mm1',$ $\overline{6}$ $m21',$ $\overline{6}$ $2m1',6/m'mm,6'/mm'm,6'/mmm',$ $6/mmm1',6/m'm'm']$	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	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$
$m$ $\overline{3}$ $1’(23')$ $[231',m'$ $\overline{3}$ $',m$ $\overline{3}$ $1']$	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	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$
$m$ $\overline{3}$ $m1’(43'2)[4321',$ $\overline{4}$ $3m1',$ $m'$ $\overline{3}$ $'m,m'$ $\overline{3}$ $'m',m$ $\overline{3}$ $m1']$	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	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$	$\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$

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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.