Information of spin space group $P{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}{4}_3{\phantom{A}}_{\phantom{}}^{\phantom{\text{-}1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\text{-}1}{2}_1{\phantom{A}}_{\phantom{}}^{\phantom{\text{-}1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\text{-}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$ ( nSSG:78.96.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-b+1/2, a+1/2, c+3/4 | x, y, z	{ 1 ‖ 4+001 | 1/2 1/2 3/4 }
b+1/2, -a+1/2, c+1/4 | x, y, z	{ 1 ‖ 4-001 | 1/2 1/2 1/4 }
-a, -b, c+1/2 | x, y, z	{ 1 ‖ 2001 | 0 0 1/2 }
a+1/2, -b+1/2, -c+1/4 | -x, -y, -z	{ -1 ‖ 2100 | 1/2 1/2 1/4 }
-a+1/2, b+1/2, -c+3/4 | -x, -y, -z	{ -1 ‖ 2010 | 1/2 1/2 3/4 }
b, a, -c | -x, -y, -z	{ -1 ‖ 2110 | 0 }
-b, -a, -c+1/2 | -x, -y, -z	{ -1 ‖ 21-10 | 0 0 1/2 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-b+1/2, a+1/2, c+3/4 | -x, y, z	{ m ‖ 4+001 | 1/2 1/2 3/4 }
b+1/2, -a+1/2, c+1/4 | -x, y, z	{ m ‖ 4-001 | 1/2 1/2 1/4 }
-a, -b, c+1/2 | -x, y, z	{ m ‖ 2001 | 0 0 1/2 }
a+1/2, -b+1/2, -c+1/4 | x, -y, -z	{ 2 ‖ 2100 | 1/2 1/2 1/4 }
-a+1/2, b+1/2, -c+3/4 | x, -y, -z	{ 2 ‖ 2010 | 1/2 1/2 3/4 }
b, a, -c | x, -y, -z	{ 2 ‖ 2110 | 0 }
-b, -a, -c+1/2 | x, -y, -z	{ 2 ‖ 21-10 | 0 0 1/2 }

Wyckoff Positions

WP	Site symmetry	Representative
4a	$..{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(a,a,0 | 0,0,0)
8b	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(a,b,c | 0,0,z)

Spin Brillouin Zone & k-little co-groups

Wavevector-k	Little co-group
A:(1/2,1/2,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
Γ:(0,0,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
M:(1/2,1/2,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
Z:(0,0,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
R:(0,1/2,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
X:(0,1/2,0)	${\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
Λ:(0,0,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
V:(1/2,1/2,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
Δ:(0,v,0)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
S:(u,u,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
Σ:(u,u,0)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
T:(u,1/2,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
U:(0,v,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
W:(0,1/2,w)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
Y:(u,1/2,0)	${\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
B:(0,v,w)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
C:(u,u,w)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
D:(u,v,0)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
E:(u,v,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
F:(u,1/2,w)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
GP:(u,v,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$

Spin Brillouin Zone

4/mmmP

Band Representations (Magnon)

8d

k-vector	k-vector-G↑	A↑G(8)
Γ:(0,0,0)	Γ:(0,0,0)	${\Gamma }_1^S(2)\oplus {\Gamma }_2^S(2)\oplus {\Gamma }_3^S{\Gamma }_4^S(4)$
A:(1/2,1/2,1/2)	A:(1/2,1/2,1/2)	$A_1^S{A}_3^S(4)\oplus A_2^S{A}_4^S(4)$
M:(1/2,1/2,0)	M:(1/2,1/2,0)	$M_1^S(2)\oplus M_2^S(2)\oplus M_3^S{M}_4^S(4)$
R:(0,1/2,1/2)	R:(0,1/2,1/2)	$2R_1^S{R}_2^S(4)$
X:(0,1/2,0)	X:(0,1/2,0)	$2X_1^S(2)\oplus 2X_2^S(2)$
Z:(0,0,1/2)	Z:(0,0,1/2)	$Z_1^S{Z}_3^S(4)\oplus Z_2^S{Z}_4^S(4)$
Δ:(0,v,0)	Δ:(0,v,0)	$4{\Delta }_1^S(2)$
Λ:(0,0,w)	Λ:(0,0,w)	${\Lambda }_1^S(2)\oplus {\Lambda }_2^S(2)\oplus {\Lambda }_3^S(2)\oplus {\Lambda }_4^S(2)$
S:(u,u,1/2)	S:(u,u,1/2)	$2S_1^S{S}_1^S(4)$
Σ:(u,u,0)	Σ:(u,u,0)	$4{\Sigma }_1^S(2)$
T:(u,1/2,1/2)	T:(u,1/2,1/2)	$2T_1^S{T}_1^S(4)$
U:(0,v,1/2)	U:(0,v,1/2)	$2U_1^S{U}_1^S(4)$
V:(1/2,1/2,w)	V:(1/2,1/2,w)	$V_1^S(2)\oplus V_2^S(2)\oplus V_3^S(2)\oplus V_4^S(2)$
W:(0,1/2,w)	W:(0,1/2,w)	$2W_1^S(2)\oplus 2W_2^S(2)$
Y:(u,1/2,0)	Y:(u,1/2,0)	$4Y_1^S(2)$
B:(0,v,w)	B:(0,v,w)	$4B_1^S(2)$
C:(u,u,w)	C:(u,u,w)	$4C_1^S(2)$
D:(u,v,0)	D:(u,v,0)	$8D_1^S(1)$	Spin Splitting
E:(u,v,1/2)	E:(u,v,1/2)	$4E_1^S{E}_1^S(2)$	Spin Splitting
F:(u,1/2,w)	F:(u,1/2,w)	$4F_1^S(2)$
GP:(u,v,w)	GP:(u,v,w)	$8G{P}_1^S(1)$	Spin Splitting