Table 1 Selection of the noncentrosymmetric crystallographic space groups with high-symmetry points, lines, and planes hosting SP-PST

Space group		Points		Lines		Planes		Example materials
Monoclinic
4	P2₁	[010]	Z, E, D, A	[010]	ΓZ, YD, XA, EC			Hf_0.5Zr_0.5O₂²¹
7	Pc	[010]	X, E, C, A			[010]	ΓYX, ZDA	BaAs₂
Orthorhombic
19	P2₁2₁2₁	[100]	X	[100]	ΓX, \({{{\bf{TR}}}}\)			Ag₂Se⁵⁵
		[010]	Y	[010]	ΓY, UR
		[001]	Z	[001]	ΓZ, SR
33	Pna2₁	[100]	Y	[100]	ΓY, UR	[100]	ΓZY, XUS	BiInO₃²⁰
		[010]	X	[010]	ΓX, ZU	[010]	ΓZX, YTS
		[001]	T
Tetragonal
99	P4mm			[100]	ΓX, ZR	[100]	ΓZX	BiFeO₃⁵⁶
				[010]	XM, RA	[010]	XRM
				\([1\bar{1}0]\)	ΓM, ZA	\([1\bar{1}0]\)	ΓZM
Trigonal
154	P3₂21	[001]	Γ^, A^	[001]	ΓA, KH			Te³⁹
154	P3₂21	[001]	Γ^, A^	[110]	ΓK, HA			Te³⁹
Hexagonal
174	\(P\bar{6}\)	[001]	Γ^, K, H, A^	[001]	ΓM, ΓK, ΓA^*,	[001]	ΓMK, LHA	Ge₃Pb₅O₁₁⁵⁷
174	\(P\bar{6}\)	[001]	Γ^, K, H, A^	[001]	LA, KH, HA	[001]	ΓMK, LHA	Ge₃Pb₅O₁₁⁵⁷
189	\(P\bar{6}2m\)	[001]	Γ^, K^, H^, A^	[001]	ΓM, LA	[001]	ΓMK, LHA	K₃Ta₃B₂O₁₂⁵⁸
189	\(P\bar{6}2m\)	[001]	Γ^, K^, H^, A^	[210]	ML	\([1\bar{1}0]\)	ΓKA	K₃Ta₃B₂O₁₂⁵⁸
Cubic
198	P2₁3	[010]	X	[010]	ΓX			OsSi
				[001]	RM
				[111]	ΓR
216	\(F\bar{4}3m\)	[100]	W	[100]	XW	\([1\bar{1}0]\)	ΓLK	Be₅Pt
216	\(F\bar{4}3m\)	[111]	L^*	\([1\bar{1}0]\)	ΓK	\([1\bar{1}0]\)	ΓLK	Be₅Pt

Note that we use the grey groups containing time-reversal symmetry (\({1}^{{\prime} }\)), which is omitted in the space group symbols for simplicity. Normal and bold fonts denote type I and type II PST, respectively. The spin direction is indicated in square brackets and expressed in terms of the conventional reciprocal lattice vectors. The definition of the reciprocal lattice vectors and the high-symmetry k-points follows the Setyawan–Curtarolo (SC) convention⁵⁹. The symbol ^* denotes that there is a type I or type II PST in some representations, and no PST for the other representations. The last column lists materials examples known from the literature or identified in the present work. The full table is given in the Supplementary Note 4.

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