Information of spin space group $I\ce{^{\text{-}1}{2}}\ce{^{\text{-}1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}}$ ( nSSG:5.23.1.1 )

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 | -x, -y, -z	{ -1 ‖ 2100 | 0 }
-a, b, -c | -x, -y, -z	{ -1 ‖ 2010 | 0 }
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+1/2 | -x, -y, -z	{ -1 ‖ 2100 | 1/2 1/2 1/2 }
-a+1/2, b+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ 2010 | 1/2 1/2 1/2 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-a, -b, c | -x, y, z	{ m ‖ 2001 | 0 }
a, -b, -c | x, -y, -z	{ 2 ‖ 2100 | 0 }
-a, b, -c | x, -y, -z	{ 2 ‖ 2010 | 0 }
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+1/2 | x, -y, -z	{ 2 ‖ 2100 | 1/2 1/2 1/2 }
-a+1/2, b+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ 2010 | 1/2 1/2 1/2 }

Wyckoff Positions

WP	Site symmetry	Representative
2a	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
2b	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/2,0,0 | 0,0,0)
2c	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,0)
2d	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,1/2,0 | 0,0,0)
4e	$\ce{^{-1}{2}}..\ce{^{\infty m}{1}} $	(a,0,0 | 0,0,0)
4f	$\ce{^{-1}{2}}..\ce{^{\infty m}{1}} $	(a,0,1/2 | 0,0,0)
4g	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(0,b,0 | 0,0,0)
4h	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(1/2,b,0 | 0,0,0)
4i	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,c | 0,0,z)
4j	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,1/2,c | 0,0,z)
8k	$\ce{^{1}{1}}\ce{^{\infty m}{1}} $	(a,b,c | 0,0,z)

Spin Brillouin Zone & k-little co-groups

Wavevector-k	Little co-group

Spin Brillouin Zone

Band Representations (Magnon)

4i

k-vector	k-vector-G↑	A↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{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)GP_{1}^{S}(2) $
Δ:(0,v,0)	B:(v,0,0)	$(Δ)B_{1}^{S}(2) $
Λ:(0,0,w)	Λ:(0,-w,0)	$Λ_{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)GP_{1}^{S}(2) $
Σ:(u,0,0)	B:(u,0,-u)	$(Σ)B_{1}^{S}(2) $
A:(0,v,w)	GP:(v,-w,0)	$(A)GP_{1}^{S}(2) $
B:(u,0,w)	GP:(u,-w,-u)	$(B)GP_{1}^{S}(2) $
C:(u,v,0)	B:(u+v,0,-u)	$2(C)B_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u+v,-w,-u)	$2GP_{1}^{S}(1)$	Spin Splitting

4j

k-vector	k-vector-G↑	A↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{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)GP_{1}^{S}(2) $
Δ:(0,v,0)	B:(v,0,0)	$(Δ)B_{1}^{S}(2) $
Λ:(0,0,w)	Λ:(0,-w,0)	$Λ_{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)GP_{1}^{S}(2) $
Σ:(u,0,0)	B:(u,0,-u)	$(Σ)B_{1}^{S}(2) $
A:(0,v,w)	GP:(v,-w,0)	$(A)GP_{1}^{S}(2) $
B:(u,0,w)	GP:(u,-w,-u)	$(B)GP_{1}^{S}(2) $
C:(u,v,0)	B:(u+v,0,-u)	$2(C)B_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u+v,-w,-u)	$2GP_{1}^{S}(1)$	Spin Splitting

8k

k-vector	k-vector-G↑	A↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{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)	$2 (T)M_{1}^{S}(1)⊕2 (T)M_{2}^{S}(1)$	Spin Splitting
W:(-1/2,1/2,-1/2)	U:(0,1/2,1/2)	$(W)U_{1}^{S}(2)⊕(W)U_{2}^{S}(2) $
X:(0,0,-1)	Y:(0,1,0)	$(X)Y_{1}^{S}(2)⊕(X)Y_{2}^{S}(2) $
D:(u,1/2,1/2)	GP:(1/2+u,-1/2,-u)	$2(D)GP_{1}^{S}(2) $
Δ:(0,v,0)	B:(v,0,0)	$2(Δ)B_{1}^{S}(2) $
Λ:(0,0,w)	Λ:(0,-w,0)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
P:(-1/2,1/2,-1+w)	U:(0,1-w,1/2)	$2(P)U_{1}^{S}(1)⊕2(P)U_{2}^{S}(1)$	Spin Splitting
Q:(1/2,v,1/2)	GP:(1/2+v,-1/2,-1/2)	$2(Q)GP_{1}^{S}(2) $
Σ:(u,0,0)	B:(u,0,-u)	$2(Σ)B_{1}^{S}(2) $
A:(0,v,w)	GP:(v,-w,0)	$2(A)GP_{1}^{S}(2) $
B:(u,0,w)	GP:(u,-w,-u)	$2(B)GP_{1}^{S}(2) $
C:(u,v,0)	B:(u+v,0,-u)	$4(C)B_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u+v,-w,-u)	$4GP_{1}^{S}(1)$	Spin Splitting