Find Spin Group - Program for Identifying Spin Space Group

File Name	Spin Space Group	G0 Symbol	L0 Symbol	it	ik
$0.607 / R u O 2$	$P^{- 1} {4_{2} /}^{1} m^{- 1} n^{1} m^{\infty m} 1$	$P 4_{2} / m n m (136)$	$C m m m (65)$	$2$	$1$

Space Group	Magnetic Space Group	Spin Part PG	Configuration	Magnetic Phase
$P 4_{2} / m n m (136)$	$\begin{aligned} P 4_{2}^{'} / m n m^{'} (136.499) \\ T y p e I I I \end{aligned}$	$\begin{aligned} C i \times C_{\infty} v & = D_{\infty} h \\ - 1 \times \infty m & = \infty / m m \end{aligned}$	$C o l l i n e a r$	$A F M (A l t e r m a g n e t)$

Little Cogroup at General Position		Spin Splitting
without SOC	with SOC	without SOC	with SOC
$^{m_{100}} {- 1}^{\infty} 1$	$1$	$k - d e p e n d e n t$	$Y e s$

Effective Magnetic Point Group	Anomalous Hall Effect
$4 / m m m 1^{'}$	without SOC	with SOC
$4 / m m m 1^{'}$	$N o$	$N o$

Effective MPG Constraint on rank-3 Tensor
		without SOC	with SOC
T-odd	Quantum metric dipole	$O n l y z e r o s$	$O n l y z e r o s$
T-odd	Inversed mass dipole	$O n l y z e r o s$	$O n l y z e r o s$
T-even	Berry curvature dipole	$O n l y z e r o s$	$O n l y z e r o s$

G0-std Unit Magnetic Cell:

a	b	c	alpha	beta	gamma
4.492	4.492	3.1061	90.0	90.0	90.0

Fractional Coordinates & Moments (in G0-std Unit Magnetic Cell)

No.	Atom	Site	Moment (in lattice)
1	Ru	[0.5, 0.5, 0.5]	[-0.0, 0.0, 0.05]
2	Ru	[0.0, 0.0, 0.0]	[0.0, -0.0, -0.05]
3	O	[0.8056, 0.8056, 0.5]	[0.0, 0.0, 0.0]
4	O	[0.3056, 0.6944, 0.0]	[0.0, 0.0, 0.0]
5	O	[0.6944, 0.3056, 0.0]	[0.0, 0.0, 0.0]
6	O	[0.1944, 0.1944, 0.5]	[0.0, 0.0, 0.0]

Elements of the Spin Space Group (G0-std-setting):

Real Space Elements are under the basis of the lattice given above.

Spin Part Elements are represented corresponding to the SpinPartPG.

No.	Spin Rotation	Space Rotation	Space Translation	Seitz Symbol
1	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| 1 \| τ_{(0.0, 0.0, 0.0)}}$
2	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{100} \| \| 1 \| τ_{(0.0, 0.0, 0.0)}}$
3	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| m_{001} \| τ_{(0.0, 0.0, 0.0)}}$
4	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{100} \| \| m_{001} \| τ_{(0.0, 0.0, 0.0)}}$
5	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${- 1 \| \| m_{010} \| τ_{(0.5, 0.5, 0.5)}}$
6	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${2_{100} \| \| m_{010} \| τ_{(0.5, 0.5, 0.5)}}$
7	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${- 1 \| \| 2_{100} \| τ_{(0.5, 0.5, 0.5)}}$
8	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${2_{100} \| \| 2_{100} \| τ_{(0.5, 0.5, 0.5)}}$
9	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| m_{1 - 10} \| τ_{(0.0, 0.0, 0.0)}}$
10	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{100} \| \| m_{1 - 10} \| τ_{(0.0, 0.0, 0.0)}}$
11	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| 2_{110} \| τ_{(0.0, 0.0, 0.0)}}$
12	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{100} \| \| 2_{110} \| τ_{(0.0, 0.0, 0.0)}}$
13	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${- 1 \| \| 4_{001}^{3} \| τ_{(0.5, 0.5, 0.5)}}$
14	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${2_{100} \| \| 4_{001}^{3} \| τ_{(0.5, 0.5, 0.5)}}$
15	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${- 1 \| \| - 4_{001}^{1} \| τ_{(0.5, 0.5, 0.5)}}$
16	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${2_{100} \| \| - 4_{001}^{1} \| τ_{(0.5, 0.5, 0.5)}}$
17	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${- 1 \| \| 4_{001}^{1} \| τ_{(0.5, 0.5, 0.5)}}$
18	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${2_{100} \| \| 4_{001}^{1} \| τ_{(0.5, 0.5, 0.5)}}$
19	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${- 1 \| \| - 4_{001}^{3} \| τ_{(0.5, 0.5, 0.5)}}$
20	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${2_{100} \| \| - 4_{001}^{3} \| τ_{(0.5, 0.5, 0.5)}}$
21	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| m_{110} \| τ_{(0.0, 0.0, 0.0)}}$
22	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{100} \| \| m_{110} \| τ_{(0.0, 0.0, 0.0)}}$
23	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| 2_{1 - 10} \| τ_{(0.0, 0.0, 0.0)}}$
24	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{100} \| \| 2_{1 - 10} \| τ_{(0.0, 0.0, 0.0)}}$
25	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${- 1 \| \| m_{100} \| τ_{(0.5, 0.5, 0.5)}}$
26	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${2_{100} \| \| m_{100} \| τ_{(0.5, 0.5, 0.5)}}$
27	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${- 1 \| \| 2_{010} \| τ_{(0.5, 0.5, 0.5)}}$
28	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${2_{100} \| \| 2_{010} \| τ_{(0.5, 0.5, 0.5)}}$
29	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| 2_{001} \| τ_{(0.0, 0.0, 0.0)}}$
30	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{100} \| \| 2_{001} \| τ_{(0.0, 0.0, 0.0)}}$
31	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| - 1 \| τ_{(0.0, 0.0, 0.0)}}$
32	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{100} \| \| - 1 \| τ_{(0.0, 0.0, 0.0)}}$

FINDSPINGROUP: Identify Spin Group and Related Properties