Typesetting…

Crystal and Magnetic Structure

FormulaCoTa3S6
Cell\(a\)\({} = 11.4800\,\mathring{\mathrm{A}}\)\(\alpha\)\({} = 90.00^\circ\)\(b\)\({} = 11.4800\,\mathring{\mathrm{A}}\)\(\beta\)\({} = 90.00^\circ\)\(c\)\({} = 11.9300\,\mathring{\mathrm{A}}\)\(\gamma\)\({} = 120.00^\circ\)
Sites80 in the cell, 8 magnetic
Crystal SG\(P6_{3}22\) (#182)
MSG Family SG\(P321\) (#150)
MSG (BNS)\(P32'1\) (#150.27)
ReferencesNat. Commun. 14, 8346 (2023)
Nat. Phys. 19, 961 (2023)
All values derived from the MCIF.

Spin-Space Symmetry

Crystal SG (\(G_0\))\(P6_{3}22\) (#182)
\(\vec{Q}\) in \(G_0\)\(\{M\,(0,\tfrac{1}{2},0)\}\)
Parent SG (\(G\))\(P6_{3}22\) (#182)
Point Group (\(P\))\(D_{6}\) (\(622\))\(622\) (\(D_{6}\))
RegularYes, \([G_0 : G] = 1\)
\(\dim\Gamma\)3, non-coplanar
\(\Lambda^1\Gamma = \Gamma\)\(M_{1}\), \(\{M\,(0,\tfrac{1}{2},0),\) \(M\,(\tfrac{1}{2},0,0),\) \(M\,(\tfrac{1}{2},-\tfrac{1}{2},0)\}\)
\(\Lambda^2\Gamma = \Gamma\otimes\det\Gamma\)\(M_{2}\), \(\{M\,(0,\tfrac{1}{2},0),\) \(M\,(\tfrac{1}{2},0,0),\) \(M\,(\tfrac{1}{2},-\tfrac{1}{2},0)\}\)
\(\Lambda^3\Gamma = \det\Gamma\)\(A_{2}\), \(\{\vec{0}\}\)

Stars of the wavevectors \(\vec{k}\) of each irreducible part, in order, in the setting of \(G\). (Spin space group 182.4.4.2)

Magnetic Symmetry

Crystal SG (\(G_0\))\(P6_{3}22\) (#182)
\(\vec{Q}\) in \(G_0\)\(\{M\,(0,\tfrac{1}{2},0)\}\)
Parent SG (\(G\))\(P321\) (#150)
Point Group (\(P\))\(D_{3}\) (\(32\))\(32\) (\(D_{3}\))
RegularNo, \([G_0 : G] = 8\)
\(\Phi\)\(A_{2}\), \(\{\vec{0}\}\)
\(\Lambda^1\Gamma = \Phi\otimes\Gamma_1^+\)\(A_{1} \oplus E\), \(\{\vec{0}\}\), \(\{\vec{0}\}\)
\(\Lambda^2\Gamma = \Gamma_1^+\)\(A_{2} \oplus E\), \(\{\vec{0}\}\), \(\{\vec{0}\}\)
\(\Lambda^3\Gamma = \Phi\)\(A_{2}\), \(\{\vec{0}\}\)

Stars of the wavevectors \(\vec{k}\) of each irreducible part, in order, in the setting of \(G\).

Non-Relativistic Responses

ResponseSymbolRequired OverlapAllowed
Net Spin Magnetization\(\vec{M}\)\(\Gamma\cap\Gamma_0^+\) ✗
Piezomagnetism\(\vec{C}_{\mu\nu}\)\(\Gamma\cap\Gamma_2^+\) ✗
Time-Odd Spin Conductivity\(\Sigma^{-}\)\(\Gamma\cap(\Gamma_0^+\oplus\Gamma_1^+\oplus\Gamma_2^+)\) ✗
Time-Even Spin Conductivity\(\Sigma^{+}\)\(\Lambda^2\Gamma\cap(\Gamma_0^+\oplus\Gamma_1^+\oplus\Gamma_2^+)\) ✗
Anomalous Hall Conductivity\(\sigma^{-}\)\(\Lambda^3\Gamma\cap\Gamma_1^+\) ✓
Time-Odd Edelstein Effect\(\alpha^{-}\)\(\Gamma\cap\Gamma_1^-\) ✗
Time-Even Edelstein Effect\(\alpha^{+}\)\(\Lambda^2\Gamma\cap\Gamma_1^-\) ✗
Spin Texture, Even Rank\(\vec{s}(\vec{q})\)\(\Gamma\cap\Gamma_\ell^+\) ✗a
Spin Texture, Odd Rank\(\vec{s}(\vec{q})\)\(\Lambda^2\Gamma\cap\Gamma_\ell^-\) ✗a

\(\vec{M}\) = net spin magnetization, \(\vec{C}_{\mu\nu}\) = piezomagnetism, \(\Sigma^{-}\) = time-odd spin conductivity, \(\Sigma^{+}\) = time-even spin conductivity, \(\sigma^{-}\) = anomalous Hall conductivity, \(\alpha^{-}\) = time-odd Edelstein effect, \(\alpha^{+}\) = time-even Edelstein effect, \(\vec{s}(\vec{q})\) = spin texture

a Lowest multipole rank \(\ell\) with the required overlap: even ranks from \(\Lambda^1\Gamma\), odd ranks from \(\Lambda^2\Gamma\).

Relativistic Responses

ResponseSymbolRequired OverlapAllowed
Net Spin Magnetization\(\vec{M}\)\(\Gamma\cap\Gamma_0^+\) ✓
Piezomagnetism\(\vec{C}_{\mu\nu}\)\(\Gamma\cap\Gamma_2^+\) ✓a
Time-Odd Spin Conductivity\(\Sigma^{-}\)\(\Gamma\cap(\Gamma_0^+\oplus\Gamma_1^+\oplus\Gamma_2^+)\) ✓
Time-Even Spin Conductivity\(\Sigma^{+}\)\(\Lambda^2\Gamma\cap(\Gamma_0^+\oplus\Gamma_1^+\oplus\Gamma_2^+)\) ✓
Anomalous Hall Conductivity\(\sigma^{-}\)\(\Lambda^3\Gamma\cap\Gamma_1^+\) ✓
Time-Odd Edelstein Effect\(\alpha^{-}\)\(\Gamma\cap\Gamma_1^-\) ✓
Time-Even Edelstein Effect\(\alpha^{+}\)\(\Lambda^2\Gamma\cap\Gamma_1^-\) ✓
Spin Texture, Even Rank\(\vec{s}(\vec{q})\)\(\Gamma\cap\Gamma_\ell^+\) s-wave \((\ell=0)\)b
Spin Texture, Odd Rank\(\vec{s}(\vec{q})\)\(\Lambda^2\Gamma\cap\Gamma_\ell^-\) p-wave \((\ell=1)\)b

\(\vec{M}\) = net spin magnetization, \(\vec{C}_{\mu\nu}\) = piezomagnetism, \(\Sigma^{-}\) = time-odd spin conductivity, \(\Sigma^{+}\) = time-even spin conductivity, \(\sigma^{-}\) = anomalous Hall conductivity, \(\alpha^{-}\) = time-odd Edelstein effect, \(\alpha^{+}\) = time-even Edelstein effect, \(\vec{s}(\vec{q})\) = spin texture

a The order is uncompensated, so \(\vec{M}\) is already allowed.
b Lowest multipole rank \(\ell\) with the required overlap: even ranks from \(\Lambda^1\Gamma\), odd ranks from \(\Lambda^2\Gamma\).

Multipoles, \(\boldsymbol{P = D_{6}\ (622)}\)\(\boldsymbol{P = 622\ (D_{6})}\)

MultipoleMultipoleDecomposition
Scalar\(\Gamma_0^+\)\(A_{1}\)
Polar Vector\(\Gamma_1^-\)\(A_{2}\) \(\oplus\) \(E_{1}\)
Axial Vector\(\Gamma_1^+\)\(A_{2}\) \(\oplus\) \(E_{1}\)
Quadrupole\(\Gamma_2^+\)\(A_{1}\) \(\oplus\) \(E_{1}\) \(\oplus\) \(E_{2}\)
MultipoleMultipoleDecomposition
Octupole\(\Gamma_3^-\)\(A_{2}\) \(\oplus\) \(B_{1}\) \(\oplus\) \(B_{2}\) \(\oplus\) \(E_{1}\) \(\oplus\) \(E_{2}\)
Hexadecapole\(\Gamma_4^+\)\(A_{1}\) \(\oplus\) \(B_{1}\) \(\oplus\) \(B_{2}\) \(\oplus\) \(E_{1}\) \(\oplus\) \(2E_{2}\)
Dotriacontapole\(\Gamma_5^-\)\(A_{2}\) \(\oplus\) \(B_{1}\) \(\oplus\) \(B_{2}\) \(\oplus\) \(2E_{1}\) \(\oplus\) \(2E_{2}\)
Hexacontatetrapole\(\Gamma_6^+\)\(2A_{1}\) \(\oplus\) \(A_{2}\) \(\oplus\) \(B_{1}\) \(\oplus\) \(B_{2}\) \(\oplus\) \(2E_{1}\) \(\oplus\) \(2E_{2}\)

\(\Gamma_0^+\) = scalar, \(\Gamma_1^-\) = polar vector, \(\Gamma_1^+\) = axial vector, \(\Gamma_2^+\) = quadrupole, \(\Gamma_3^-\) = octupole, \(\Gamma_4^+\) = hexadecapole, \(\Gamma_5^-\) = dotriacontapole, \(\Gamma_6^+\) = hexacontatetrapole

Representation

Each box is one operation \(\ssop{\mat{\Gamma}(g)}{g}\): \(\ssop{U}{g}\)\(\{\mat{\Gamma}(g)\,\Vert\,g\}\): \(\{U\,\Vert\,g\}\), the space-group operation \(g\) with its spin rotation \(U\) named, above the matrix \(\mat{\Gamma}(g)\) in the spin basis \(\vhat{N}_i\) below or, with the toggle, the three-dimensional representative \(\mat{U}\) in Cartesian spin coordinates.

Spin Frame

Symmetry-adapted basis of the spin subspace (Cartesian components): \(\vhat{N}_1 = (\tfrac{\sqrt{2}}{2},\, \tfrac{\sqrt{6}}{6},\, \tfrac{\sqrt{3}}{3})\), \(\vhat{N}_2 = (0,\, \tfrac{\sqrt{6}}{3},\, -\tfrac{\sqrt{3}}{3})\), \(\vhat{N}_3 = (-\tfrac{\sqrt{2}}{2},\, \tfrac{\sqrt{6}}{6},\, \tfrac{\sqrt{3}}{3})\). In this basis translations act diagonally wherever their wavevectors are time-reversal invariant.

Lattice Translations and Centerings

\([C_{2}\,\Vert\,1\,|\,1\,0\,0]\)\(\{2\,\Vert\,1\,|\,1\,0\,0\}\)
\(\begin{pmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{pmatrix}\)
\(\begin{pmatrix}-1 & 0 & 0 \\ 0 & \tfrac{1}{3} & -\tfrac{2\sqrt{2}}{3} \\ 0 & -\tfrac{2\sqrt{2}}{3} & -\tfrac{1}{3}\end{pmatrix}\)
\([C_{2}\,\Vert\,1\,|\,0\,1\,0]\)\(\{2\,\Vert\,1\,|\,0\,1\,0\}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{pmatrix}\)
\(\begin{pmatrix}0 & \tfrac{\sqrt{3}}{3} & \tfrac{\sqrt{6}}{3} \\ \tfrac{\sqrt{3}}{3} & -\tfrac{2}{3} & \tfrac{\sqrt{2}}{3} \\ \tfrac{\sqrt{6}}{3} & \tfrac{\sqrt{2}}{3} & -\tfrac{1}{3}\end{pmatrix}\)
\([1\,\Vert\,1\,|\,0\,0\,1]\)\(\{1\,\Vert\,1\,|\,0\,0\,1\}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)

Point Operations

One coset representative \(g=[\mat{R}\,|\,\vec{t}]\)\(g=\{\mat{R}\,|\,\vec{t}\}\) per element of \(P\), in the conventional setting of \(G\). Operations marked generate \(P\).

\([1\,\Vert\,1\,|\,\vec{0}]\)\(\{1\,\Vert\,1\,|\,0\}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\([1\,\Vert\,C_{2z}\,|\,0\,0\,\tfrac{1}{2}]\)\(\{1\,\Vert\,2_{001}\,|\,0\,0\,\tfrac{1}{2}\}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\([C_{3}^{+}\,\Vert\,C_{6z}^{+}\,|\,0\,0\,\tfrac{1}{2}]\)\(\{3^{+}\,\Vert\,6^{+}_{001}\,|\,0\,0\,\tfrac{1}{2}\}\)
\(\begin{pmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{pmatrix}\)
\(\begin{pmatrix}-\tfrac{1}{2} & -\tfrac{\sqrt{3}}{6} & \tfrac{\sqrt{6}}{3} \\ \tfrac{\sqrt{3}}{6} & \tfrac{5}{6} & \tfrac{\sqrt{2}}{3} \\ -\tfrac{\sqrt{6}}{3} & \tfrac{\sqrt{2}}{3} & -\tfrac{1}{3}\end{pmatrix}\)
\([C_{3}^{-}\,\Vert\,C_{6z}^{-}\,|\,0\,0\,\tfrac{1}{2}]\)\(\{3^{-}\,\Vert\,6^{-}_{001}\,|\,0\,0\,\tfrac{1}{2}\}\)
\(\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{pmatrix}\)
\(\begin{pmatrix}-\tfrac{1}{2} & \tfrac{\sqrt{3}}{6} & -\tfrac{\sqrt{6}}{3} \\ -\tfrac{\sqrt{3}}{6} & \tfrac{5}{6} & \tfrac{\sqrt{2}}{3} \\ \tfrac{\sqrt{6}}{3} & \tfrac{\sqrt{2}}{3} & -\tfrac{1}{3}\end{pmatrix}\)
\([C_{3}^{-}\,\Vert\,C_{3z}^{+}\,|\,\vec{0}]\)\(\{3^{-}\,\Vert\,3^{+}_{001}\,|\,0\}\)
\(\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{pmatrix}\)
\(\begin{pmatrix}-\tfrac{1}{2} & \tfrac{\sqrt{3}}{6} & -\tfrac{\sqrt{6}}{3} \\ -\tfrac{\sqrt{3}}{6} & \tfrac{5}{6} & \tfrac{\sqrt{2}}{3} \\ \tfrac{\sqrt{6}}{3} & \tfrac{\sqrt{2}}{3} & -\tfrac{1}{3}\end{pmatrix}\)
\([C_{3}^{+}\,\Vert\,C_{3z}^{-}\,|\,\vec{0}]\)\(\{3^{+}\,\Vert\,3^{-}_{001}\,|\,0\}\)
\(\begin{pmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{pmatrix}\)
\(\begin{pmatrix}-\tfrac{1}{2} & -\tfrac{\sqrt{3}}{6} & \tfrac{\sqrt{6}}{3} \\ \tfrac{\sqrt{3}}{6} & \tfrac{5}{6} & \tfrac{\sqrt{2}}{3} \\ -\tfrac{\sqrt{6}}{3} & \tfrac{\sqrt{2}}{3} & -\tfrac{1}{3}\end{pmatrix}\)
\([\sigma_{x}\,\Vert\,C_{2[100]}\,|\,\vec{0}]\)\(\{m_{x}\,\Vert\,2_{100}\,|\,0\}\)
\(\begin{pmatrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{pmatrix}\)
\(\begin{pmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\([\sigma\,\Vert\,C_{2[010]}\,|\,\vec{0}]\)\(\{m\,\Vert\,2_{010}\,|\,0\}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{pmatrix}\)
\(\begin{pmatrix}\tfrac{1}{2} & -\tfrac{\sqrt{3}}{6} & \tfrac{\sqrt{6}}{3} \\ -\tfrac{\sqrt{3}}{6} & \tfrac{5}{6} & \tfrac{\sqrt{2}}{3} \\ \tfrac{\sqrt{6}}{3} & \tfrac{\sqrt{2}}{3} & -\tfrac{1}{3}\end{pmatrix}\)
\([\sigma\,\Vert\,C_{2[110]}\,|\,\vec{0}]\)\(\{m\,\Vert\,2_{110}\,|\,0\}\)
\(\begin{pmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\(\begin{pmatrix}\tfrac{1}{2} & \tfrac{\sqrt{3}}{6} & -\tfrac{\sqrt{6}}{3} \\ \tfrac{\sqrt{3}}{6} & \tfrac{5}{6} & \tfrac{\sqrt{2}}{3} \\ -\tfrac{\sqrt{6}}{3} & \tfrac{\sqrt{2}}{3} & -\tfrac{1}{3}\end{pmatrix}\)
\([\sigma\,\Vert\,C_{2[1\bar{1}0]}\,|\,0\,0\,\tfrac{1}{2}]\)\(\{m\,\Vert\,2_{1\bar{1}0}\,|\,0\,0\,\tfrac{1}{2}\}\)
\(\begin{pmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\(\begin{pmatrix}\tfrac{1}{2} & \tfrac{\sqrt{3}}{6} & -\tfrac{\sqrt{6}}{3} \\ \tfrac{\sqrt{3}}{6} & \tfrac{5}{6} & \tfrac{\sqrt{2}}{3} \\ -\tfrac{\sqrt{6}}{3} & \tfrac{\sqrt{2}}{3} & -\tfrac{1}{3}\end{pmatrix}\)
\([\sigma\,\Vert\,C_{2[210]}\,|\,0\,0\,\tfrac{1}{2}]\)\(\{m\,\Vert\,2_{210}\,|\,0\,0\,\tfrac{1}{2}\}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{pmatrix}\)
\(\begin{pmatrix}\tfrac{1}{2} & -\tfrac{\sqrt{3}}{6} & \tfrac{\sqrt{6}}{3} \\ -\tfrac{\sqrt{3}}{6} & \tfrac{5}{6} & \tfrac{\sqrt{2}}{3} \\ \tfrac{\sqrt{6}}{3} & \tfrac{\sqrt{2}}{3} & -\tfrac{1}{3}\end{pmatrix}\)
\([\sigma_{x}\,\Vert\,C_{2[120]}\,|\,0\,0\,\tfrac{1}{2}]\)\(\{m_{x}\,\Vert\,2_{120}\,|\,0\,0\,\tfrac{1}{2}\}\)
\(\begin{pmatrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{pmatrix}\)
\(\begin{pmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)

Multipoles, \(\boldsymbol{P = D_{3}\ (32)}\)\(\boldsymbol{P = 32\ (D_{3})}\)

MultipoleMultipoleDecomposition
Scalar\(\Gamma_0^+\)\(A_{1}\)
Polar Vector\(\Gamma_1^-\)\(A_{2}\) \(\oplus\) \(E\)
Axial Vector\(\Gamma_1^+\)\(A_{2}\) \(\oplus\) \(E\)
Quadrupole\(\Gamma_2^+\)\(A_{1}\) \(\oplus\) \(2E\)
MultipoleMultipoleDecomposition
Octupole\(\Gamma_3^-\)\(A_{1}\) \(\oplus\) \(2A_{2}\) \(\oplus\) \(2E\)
Hexadecapole\(\Gamma_4^+\)\(2A_{1}\) \(\oplus\) \(A_{2}\) \(\oplus\) \(3E\)
Dotriacontapole\(\Gamma_5^-\)\(A_{1}\) \(\oplus\) \(2A_{2}\) \(\oplus\) \(4E\)
Hexacontatetrapole\(\Gamma_6^+\)\(3A_{1}\) \(\oplus\) \(2A_{2}\) \(\oplus\) \(4E\)

\(\Gamma_0^+\) = scalar, \(\Gamma_1^-\) = polar vector, \(\Gamma_1^+\) = axial vector, \(\Gamma_2^+\) = quadrupole, \(\Gamma_3^-\) = octupole, \(\Gamma_4^+\) = hexadecapole, \(\Gamma_5^-\) = dotriacontapole, \(\Gamma_6^+\) = hexacontatetrapole

Representation

Each box is one operation \(\ssop{\mat{\Gamma}(g)}{g}\): \(\ssop{U}{g}\)\(\{\mat{\Gamma}(g)\,\Vert\,g\}\): \(\{U\,\Vert\,g\}\), the space-group operation \(g\) with its spin rotation \(U\) named, above the matrix \(\mat{\Gamma}(g)\) in the spin basis \(\vhat{N}_i\) below or, with the toggle, the three-dimensional representative \(\mat{U}\) in Cartesian spin coordinates. The basis is ordered as \(\Gamma = A_{1} \oplus E\), whose parts have dimensions \(\dim A_{1} = 1\) and \(\dim E = 2\).

Spin Frame

Symmetry-adapted basis of the spin subspace (Cartesian components): \(\vhat{N}_1 = (0,\, 0,\, 1)\), \(\vhat{N}_2 = (1,\, 0,\, 0)\), \(\vhat{N}_3 = (0,\, 1,\, 0)\). In this basis translations act diagonally wherever their wavevectors are time-reversal invariant.

Lattice Translations and Centerings

\([1\,\Vert\,1\,|\,1\,0\,0]\)\(\{1\,\Vert\,1\,|\,1\,0\,0\}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\([1\,\Vert\,1\,|\,0\,1\,0]\)\(\{1\,\Vert\,1\,|\,0\,1\,0\}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\([1\,\Vert\,1\,|\,0\,0\,1]\)\(\{1\,\Vert\,1\,|\,0\,0\,1\}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)

Point Operations

One coset representative \(g=[\mat{R}\,|\,\vec{t}]\)\(g=\{\mat{R}\,|\,\vec{t}\}\) per element of \(P\), in the conventional setting of \(G\). Operations marked generate \(P\).

\([1\,\Vert\,1\,|\,\vec{0}]\)\(\{1\,\Vert\,1\,|\,0\}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\([C_{3z}^{+}\,\Vert\,C_{3z}^{+}\,|\,\vec{0}]\)\(\{3^{+}_{z}\,\Vert\,3^{+}_{001}\,|\,0\}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & -\tfrac{1}{2} & -\tfrac{\sqrt{3}}{2} \\ 0 & \tfrac{\sqrt{3}}{2} & -\tfrac{1}{2}\end{pmatrix}\)
\(\begin{pmatrix}-\tfrac{1}{2} & -\tfrac{\sqrt{3}}{2} & 0 \\ \tfrac{\sqrt{3}}{2} & -\tfrac{1}{2} & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\([C_{3z}^{-}\,\Vert\,C_{3z}^{-}\,|\,\vec{0}]\)\(\{3^{-}_{z}\,\Vert\,3^{-}_{001}\,|\,0\}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & -\tfrac{1}{2} & \tfrac{\sqrt{3}}{2} \\ 0 & -\tfrac{\sqrt{3}}{2} & -\tfrac{1}{2}\end{pmatrix}\)
\(\begin{pmatrix}-\tfrac{1}{2} & \tfrac{\sqrt{3}}{2} & 0 \\ -\tfrac{\sqrt{3}}{2} & -\tfrac{1}{2} & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\([\sigma\,\Vert\,C_{2[010]}\,|\,\vec{0}]\)\(\{m\,\Vert\,2_{010}\,|\,0\}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & \tfrac{1}{2} & \tfrac{\sqrt{3}}{2} \\ 0 & \tfrac{\sqrt{3}}{2} & -\tfrac{1}{2}\end{pmatrix}\)
\(\begin{pmatrix}\tfrac{1}{2} & \tfrac{\sqrt{3}}{2} & 0 \\ \tfrac{\sqrt{3}}{2} & -\tfrac{1}{2} & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\([\sigma_{x}\,\Vert\,C_{2[100]}\,|\,\vec{0}]\)\(\{m_{x}\,\Vert\,2_{100}\,|\,0\}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\(\begin{pmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\)
\([\sigma\,\Vert\,C_{2[110]}\,|\,\vec{0}]\)\(\{m\,\Vert\,2_{110}\,|\,0\}\)
\(\begin{pmatrix}1 & 0 & 0 \\ 0 & \tfrac{1}{2} & -\tfrac{\sqrt{3}}{2} \\ 0 & -\tfrac{\sqrt{3}}{2} & -\tfrac{1}{2}\end{pmatrix}\)
\(\begin{pmatrix}\tfrac{1}{2} & -\tfrac{\sqrt{3}}{2} & 0 \\ -\tfrac{\sqrt{3}}{2} & -\tfrac{1}{2} & 0 \\ 0 & 0 & 1\end{pmatrix}\)

Sites

Crystal Sites

One row per site of the crystal: its Wyckoff position and site symmetry in \(G_0\).

SiteWyckoffSite SymmetryFractional
Ta14f\(C_{3}\) (\(3..\))\(3..\) (\(C_{3}\))0.1667,0.3333,0.5009
Ta22a\(D_{3}\) (\(32.\))\(32.\) (\(D_{3}\))0.0000,0.0000,0.0000
Co12d\(D_{3}\) (\(3.2\))\(3.2\) (\(D_{3}\))0.1667,0.3333,0.2500
S112i\(C_{1}\) (\(1\))\(1\) (\(C_{1}\))0.1673,0.0003,0.1316

Magnetic Sites

Every magnetic site in the cell, with its moment in Cartesian coordinates.

SiteFractionalm [μB]|m| [μB]Allowed
Co10.1667,0.3333,0.2500+0.000,+0.000,+1.2701.270(0, 0, mz)
Co20.8333,0.6667,0.7500+0.000,+0.000,+1.2701.270(0, 0, mz)
Co30.1667,0.8333,0.2500+1.037,+0.599,−0.4231.270(mx, my, mz)
Co40.3333,0.1667,0.7500−1.037,+0.599,−0.4231.270(−mx+my, my, mz)
Co50.6667,0.3333,0.2500+0.000,−1.197,−0.4231.270(−mx+my, −mx, mz)
Co60.8333,0.1667,0.7500+1.037,+0.599,−0.4231.270(mx, mx−my, mz)
Co70.6667,0.8333,0.2500−1.037,+0.599,−0.4231.270(−my, mx−my, mz)
Co80.3333,0.6667,0.7500+0.000,−1.197,−0.4231.270(−my, −mx, mz)
Net+0.000,+0.002,+0.0020.003

Verification
48 operations from spinspg map the structure onto itselfpositions \(1.3\times10^{-11}\) Å, moments \(7.4\times10^{-13}\)
Spin operations preserve the moment subspace \(V\)\(0\)
Spin-only partners agree on \(V\) (\(\Gamma\) well defined)\(0\)
\(\Gamma\) orthogonal / homomorphism\(1.8\times10^{-15}\) / \(1.5\times10^{-15}\)
Cross-check with FindSpinGroup: same \(G_0, G\), order✓
 same parent group as Cartesian operations✓
 same representatives \(U\)\(\max|\Delta U| = 5.6\times10^{-13}\)
 same content of \(\Gamma\) and same responses✓
 FindSpinGroup SSG label (reference only)182.4.4.2
Tolerancespositions 0.01 Å, moments 0.01 \(\mu_\mathrm{B}\)

Independent Checks

FindSpinGroup, without spin–orbit coupling✓ 13 agree·≈ 1 known
≈ Anomalous Hall \(\sigma^{-}\) (ahc_wo_soc as returned): Find Γ allowed, FindSpinGroup No. FindSpinGroup issue: is_ahc returns No for every group (its lookup misses); its effective magnetic point group is ferromagnetic
Neumann's principle, non-relativistic✓ 10 agree
FindSpinGroup, with spin–orbit coupling✓ 15 agree
Neumann's principle, relativistic✓ 10 agree
Magnetic space group: spglib, FindSpinGroup and the MCIF✓ 9 agree
Point group and \(\Phi\): FindSpinGroup✓ 5 agree
The paper's rules for magnetic space groups✓ 3 agree

Notes

Notation follows Spin Space Symmetry Without Spin Space Groups. The spin space group is encoded by the parent space group \(G\) and the real representation \(\Gamma\) of \(G\) on the span \(V\) of the moments, \(\mat{\Gamma}(g)=\trp{\mat{Q}}\mat{S}\mat{Q}\) for each spin-space operation \(\ssop{\mat{S}}{g}\); its three-dimensional representatives are \(\mat{U}=\mat{\Gamma}(g)\mat{1}\) (collinear), \(\operatorname{diag}(\mat{\Gamma}(g),\det\mat{\Gamma}(g))\) (coplanar) and \(\mat{\Gamma}(g)\) (non-coplanar). A response is allowed when the \(\vec{k}=\vec{0}\) part of the indicated exterior power overlaps the indicated point-group representation (\(\Gamma_\ell^\pm\): rank-\(\ell\) multipole of parity ±). Mulliken labels follow the Bilbao conventions (\(C_2'\) along the conventional \(a\) axes, \(\sigma_v\) containing them). Parts at finite \(\vec{k}\) carry Miller–Love labels (ISO-IR tables) in the standard setting of \(G\) nearest the MCIF's standard setting of \(G_0\) (origin at \((0, 0, 0)\) in it). Wavevectors are in the conventional reciprocal basis of the group named, in its standard setting (\(G\) for \(\Gamma\) and its exterior powers, \(G_0\) for the propagation vector \(\vec{Q}\)): components \(\vec{k}=(k_1,k_2,k_3)\) along the reciprocal vectors \(\vec{b}_j\) of its conventional cell \(\vec{a}_i\), with \(\vec{a}_i\cdot\vec{b}_j=\delta_{ij}\), so a translation \(\vec{t}=\sum_i t_i\vec{a}_i\) carries the phase \(e^{2\pi i\,\vec{k}\cdot\vec{t}}\). Each is reduced to its shortest representative modulo that group's reciprocal lattice, with CDML labels where tabulated. Operations use Seitz notation in the standard setting chosen by spglib. Source: file 05_CoTa3S6_tetrahedral_tripleQ.mcif.