Table 1 Classification of spin texture according to the lowest order of k-polynomial of \(k\,\cdot\, p\) Hamiltonian allowed by the magnetic point groups of k-points

0

Zeeman

\(1,{\bar{\bf{1}}},2,\,{2}^{{\prime} },{m},\,{m}^{{\prime} },\bf{2/m},\bf{{2}^{{\prime} }/{m}^{{\prime} }},\) \({2}^{{\prime} }{2}^{{\prime} }2,{m}^{{\prime} }m{2}^{{\prime} },\bf{{m}^{{\prime} }{m}^{{\prime} }2},\bf{{m}^{{\prime} }{m}^{{\prime} }m},\) \(4,{\bar{\bf {4}}},{\bf{4/m}},{42}^{{\prime} }{2}^{{\prime} },\bf{4{m}^{{\prime} }{m}^{{\prime} }},\) \({\bf{4/m{m}^{\prime}{m}^{\prime}}},{{3}},{\bar{{3}}},{{32}}^{{\prime} },{{3}}{{m}}^{{\prime} },{\bar{\bf{3}}{\bf{m}}^{{\prime} }},\) \(6,{\bar{6}},{\bf{6/m}},{62}^{\prime}{2}^{\prime},{\bar{{6}}}{{m}}^{{\prime} }{{2}}^{{\prime}},\) \(\bf{6/{mm}^{\prime} m^{\prime}}\)

Break \(T\)

MoSe₂¹⁸, MnTe

1

Linear

\({11}^{{\prime} },{21}^{{\prime} },m{1}^{{\prime} },{\bf{222}},{2221}^{{\prime} },{mm}2,\) \({mm}{21}^{{\prime} },{41}^{{\prime} },{\bf{4}}^{\prime},{\bar{{4}}}{{1}}^{{\prime} },{\bar{{4}}}^{{\prime}},{\bf{422}},\) \({4221}^{{\prime} },{\bf{4}^{{\prime} }{22}^{{\prime} }},4{mm},4{mm}{1}^{{\prime} },\) \({\bf{{4}^{{\prime} }{m}^{{\prime} }m}},{\bar{\bf{4}}{\bf{2m}}},{\bar{{4}}}{{2m}}{{1}}^{{\prime} },{\bar{{4}}}^{{\prime} }{{2}}^{{\prime}}{{m}},\) \({{\bar{\bf{4}}}^{{\prime} }{\bf 2{m}}^{{\prime}}},{\bar{\bf{4}{2}}^{{\prime} }{\bf{m}}^{{\prime} }},{{31}}^{{\prime} },{{32,321}}^{{\prime} },\) \(3m,3m{1}^{{\prime} },{61}^{{\prime} },{6}^{{\prime} },{\bf{622}},{6221}^{{\prime} },\) \({6}^{{\prime} }{22}^{{\prime} },6{mm},6{mm}{1}^{{\prime} },{6}^{{\prime} }m{m}^{{\prime} },\) \({\bf{6{m}^{{\prime} }{m}^{{\prime} }},{23}},{231}^{{\prime} },{\bf{432}},4321^{{\prime} },\) \(\bf{4^{\prime} 32^{\prime}}\)

Break \(P\)

Bi/Ag⁷⁰, GaAs⁷¹, BiTeI¹⁶

2

Quadratic

\(\bf{mmm},\bf{{4}^{{\prime} }/m},\bf{4/{mmm}},\) \({\bf{4}}^{\prime}/{\bf{mm}}^{\prime}{\bf{m}},{\bar{\bf{3}}{\bf{m}},{\bar{6}}^{\prime},\bf{{6}^{\prime}/{m}}^{\prime}},\bar{6}m2,\) \({{\bar{\bf{6}}}^{{\prime} }{\bf{m}}^{{\prime}}{\bf{2}}},{\bar{6}}^{{\prime} }m{2}^{{\prime} },\bf{6/{mmm}},\) \({\bf{6}}^{\prime}/{\bf{m}}^{\prime}{\bf{mm}}^{\prime},{\bf{m}}{\bar{\bf{3}}},{\bar{\bf{4}}^{\prime}{\bf{3m}}^{\prime }},{\bf{m}}{\bar{\bf{3}}{\bf{m}}^{\prime}}\)

Break \(T\)

BaSbPt, MnTe₂⁵⁷

3

Cubic

\({\bar{6}}{1}^{\prime},{\bar{6}}m{21}^{\prime},\) \({\bar{\bf{4}}{\bf{3m}}},{\bar{{4}}}{{3m1}}^{\prime}\)

Break \(P\)

Ge₃Pb₅O₁₁¹⁷

4

Quartic

\({\bf{m}}\bar{\bf{3}}\bf{m}\)

Break \(T\)

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