Information of spin space group $I{\phantom{A}}_{\phantom{}}^{\phantom{\text{-}1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\text{-}1}\text{b}{\phantom{A}}_{\phantom{}}^{\phantom{\text{-}1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\text{-}1}\text{a}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$ ( nSSG:5.45.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-a, -b, c | x, y, z	{ 1 ‖ 2001 | 0 }
-a, b, c+1/2 | -x, -y, -z	{ -1 ‖ m100 | 0 0 1/2 }
a, -b, c+1/2 | -x, -y, -z	{ -1 ‖ m010 | 0 0 1/2 }
a+1/2, b+1/2, c+1/2 | x, y, z	{ 1 ‖ 1 | 1/2 1/2 1/2 }
-a+1/2, -b+1/2, c+1/2 | x, y, z	{ 1 ‖ 2001 | 1/2 1/2 1/2 }
-a+1/2, b+1/2, c | -x, -y, -z	{ -1 ‖ m100 | 1/2 1/2 0 }
a+1/2, -b+1/2, c | -x, -y, -z	{ -1 ‖ m010 | 1/2 1/2 0 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-a, -b, c | -x, y, z	{ m ‖ 2001 | 0 }
-a, b, c+1/2 | x, -y, -z	{ 2 ‖ m100 | 0 0 1/2 }
a, -b, c+1/2 | x, -y, -z	{ 2 ‖ m010 | 0 0 1/2 }
a+1/2, b+1/2, c+1/2 | -x, y, z	{ m ‖ 1 | 1/2 1/2 1/2 }
-a+1/2, -b+1/2, c+1/2 | -x, y, z	{ m ‖ 2001 | 1/2 1/2 1/2 }
-a+1/2, b+1/2, c | x, -y, -z	{ 2 ‖ m100 | 1/2 1/2 0 }
a+1/2, -b+1/2, c | x, -y, -z	{ 2 ‖ m010 | 1/2 1/2 0 }

Wyckoff Positions

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

Spin Brillouin Zone & k-little co-groups

Wavevector-k	Little co-group
Γ:(0,0,0)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
X:(1,1,1)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
R:(1/2,0,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
S:(0,1/2,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
T:(1/2,1/2,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
W:(1/2,1/2,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
Δ:(0,v,0)	${\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
Λ:(0,0,w)	${\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
Σ:(u,0,0)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
A:(0,v,w)	${\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
B:(u,0,w)	${\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
C:(u,v,0)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
D:(u,1/2,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
P:(1/2,1/2,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
Q:(1/2,v,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
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

Band Representations (Magnon)

4a

k-vector	k-vector-G↑	A↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	${\Gamma }_1^S(2)$
R:(-1/2,1,-1/2)	L:(1/2,1/2,1/2)	$(R)L_1^S(2)$
S:(0,1/2,-1/2)	V:(1/2,1/2,0)	$(S)V_1^S(2)$
T:(-1/2,1/2,-1)	M:(0,1,1/2)	$2(T)M_1^S(1)$	Spin Splitting
W:(-1/2,1/2,-1/2)	U:(0,1/2,1/2)	$(W)U_1^S(2)$
X:(0,0,-1)	Y:(0,1,0)	$(X)Y_1^S(2)$
D:(u,1/2,1/2)	GP:(1/2+u,-1/2,-u)	$(D)G{P}_1^S(2)$
Δ:(0,v,0)	B:(v,0,0)	$(\Delta )B_1^S(2)$
Λ:(0,0,w)	Λ:(0,-w,0)	${\Lambda }_1^S(2)$
P:(-1/2,1/2,-1+w)	U:(0,1-w,1/2)	$2(P)U_1^S(1)$	Spin Splitting
Q:(1/2,v,1/2)	GP:(1/2+v,-1/2,-1/2)	$(Q)G{P}_1^S(2)$
Σ:(u,0,0)	B:(u,0,-u)	$(\Sigma )B_1^S(2)$
A:(0,v,w)	GP:(v,-w,0)	$(A)G{P}_1^S(2)$
B:(u,0,w)	GP:(u,-w,-u)	$(B)G{P}_1^S(2)$
C:(u,v,0)	B:(u+v,0,-u)	$(C)B_1^S(2)$
GP:(u,v,w)	GP:(u+v,-w,-u)	$2G{P}_1^S(1)$	Spin Splitting

4b

k-vector	k-vector-G↑	A↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	${\Gamma }_1^S(2)$
R:(-1/2,1,-1/2)	L:(1/2,1/2,1/2)	$(R)L_1^S(2)$
S:(0,1/2,-1/2)	V:(1/2,1/2,0)	$(S)V_1^S(2)$
T:(-1/2,1/2,-1)	M:(0,1,1/2)	$2(T)M_2^S(1)$	Spin Splitting
W:(-1/2,1/2,-1/2)	U:(0,1/2,1/2)	$(W)U_2^S(2)$
X:(0,0,-1)	Y:(0,1,0)	$(X)Y_1^S(2)$
D:(u,1/2,1/2)	GP:(1/2+u,-1/2,-u)	$(D)G{P}_1^S(2)$
Δ:(0,v,0)	B:(v,0,0)	$(\Delta )B_1^S(2)$
Λ:(0,0,w)	Λ:(0,-w,0)	${\Lambda }_1^S(2)$
P:(-1/2,1/2,-1+w)	U:(0,1-w,1/2)	$2(P)U_2^S(1)$	Spin Splitting
Q:(1/2,v,1/2)	GP:(1/2+v,-1/2,-1/2)	$(Q)G{P}_1^S(2)$
Σ:(u,0,0)	B:(u,0,-u)	$(\Sigma )B_1^S(2)$
A:(0,v,w)	GP:(v,-w,0)	$(A)G{P}_1^S(2)$
B:(u,0,w)	GP:(u,-w,-u)	$(B)G{P}_1^S(2)$
C:(u,v,0)	B:(u+v,0,-u)	$(C)B_1^S(2)$
GP:(u,v,w)	GP:(u+v,-w,-u)	$2G{P}_1^S(1)$	Spin Splitting

8c

k-vector	k-vector-G↑	A↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	${\Gamma }_1^S(2)\oplus {\Gamma }_2^S(2)$
R:(-1/2,1,-1/2)	L:(1/2,1/2,1/2)	$2(R)L_1^S(2)$
S:(0,1/2,-1/2)	V:(1/2,1/2,0)	$2(S)V_1^S(2)$
T:(-1/2,1/2,-1)	M:(0,1,1/2)	$2(T)M_1^S(1)\oplus 2(T)M_2^S(1)$	Spin Splitting
W:(-1/2,1/2,-1/2)	U:(0,1/2,1/2)	$(W)U_1^S(2)\oplus (W)U_2^S(2)$
X:(0,0,-1)	Y:(0,1,0)	$(X)Y_1^S(2)\oplus (X)Y_2^S(2)$
D:(u,1/2,1/2)	GP:(1/2+u,-1/2,-u)	$2(D)G{P}_1^S(2)$
Δ:(0,v,0)	B:(v,0,0)	$2(\Delta )B_1^S(2)$
Λ:(0,0,w)	Λ:(0,-w,0)	${\Lambda }_1^S(2)\oplus {\Lambda }_2^S(2)$
P:(-1/2,1/2,-1+w)	U:(0,1-w,1/2)	$2(P)U_1^S(1)\oplus 2(P)U_2^S(1)$	Spin Splitting
Q:(1/2,v,1/2)	GP:(1/2+v,-1/2,-1/2)	$2(Q)G{P}_1^S(2)$
Σ:(u,0,0)	B:(u,0,-u)	$2(\Sigma )B_1^S(2)$
A:(0,v,w)	GP:(v,-w,0)	$2(A)G{P}_1^S(2)$
B:(u,0,w)	GP:(u,-w,-u)	$2(B)G{P}_1^S(2)$
C:(u,v,0)	B:(u+v,0,-u)	$2(C)B_1^S(2)$
GP:(u,v,w)	GP:(u+v,-w,-u)	$4G{P}_1^S(1)$	Spin Splitting