Spin representation

A spin representation is a projective representation with a factor system for spinor wave functions. Here, we give the convention to choose the factor system for spinor in spgrep.

Factor system for spinor

Symmetry operation of the first kind

The map \(\mathrm{SO}(3) \ni \mathbf{R}_{ \theta\hat{\mathbf{n}} } \mapsto \mathbf{U} ( \mathbf{R}_{ \theta\hat{\mathbf{n}} } ) := \exp \left( -\frac{i}{2}\theta \hat{\mathbf{n}} \cdot \mathbf{\sigma} \right) \in \mathrm{SU}(2)\) is not surjective, where \(\mathbf{\sigma}_{i} \, (i=x,y,z)\) are Pauli matrices

\[\begin{split} \mathbf{\sigma}_{x} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}, \mathbf{\sigma}_{y} = \begin{pmatrix} 0 & -i \\ i & 0 \\ \end{pmatrix}, \mathbf{\sigma}_{z} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}. \end{split}\]

In fact, \(\mathbf{R}_{\theta\hat{\mathbf{n}}}\) and \(\mathbf{R}_{2\pi - \theta, -\hat{\mathbf{n}}}\) represents the identical rotation. However they gives different unitary matrices with opposite signs. We choose either of the unitary matices for each rotations as convention. In particular, we choose \(\theta=0\) for identity \(\mathbf{E}\). We define a factor system from the ambiguity as

\[ \mathbf{U}(\mathbf{R}) \mathbf{U}(\mathbf{R}') =: z(\mathbf{R}, \mathbf{R}') \mathbf{U}(\mathbf{R}\mathbf{R}') \quad (\mathbf{R}, \mathbf{R}' \in \mathrm{SO}(3)). \]

In our convention, \(z(\mathbf{E}, \mathbf{R}) = z(\mathbf{R}, \mathbf{E}) = 1 \,(\forall \mathbf{R} \in \mathrm{SO}(3))\). Also, this representation matrix adapts Condon-Shortley phase.

We define the action of a symmetry operation of the first kind on spinor as

\[\begin{split} \left[ (\mathbf{R}, \mathbf{v}) \mathbf{\Psi} \right] (\mathbf{r}) &:= \mathbf{U}( \mathbf{R} ) \mathbf{\Psi}( (\mathbf{R}, \mathbf{v})^{-1} \mathbf{r}) \\ \Rightarrow \left[ g_{1} \left[ g_{2} \mathbf{\Psi} \right] \right](\mathbf{r}) &= \mathbf{U}(\mathbf{p}_{g_{1}}) \left[ g_{2} \mathbf{\Psi} \right](g_{1}^{-1}\mathbf{r}) \\ &= z(\mathbf{p}_{g_{1}}, \mathbf{p}_{g_{2}}) \left[ (g_{1}g_{2}) \mathbf{\Psi} \right](\mathbf{r}). \end{split}\]

Consider Bloch function with \(\mathbf{k}\)

\[ \mathbf{\Psi}_{\mathbf{k}}(\mathbf{r}) = e^{ i \mathbf{k} \cdot \mathbf{r} } \mathbf{u}_{\mathbf{k}}(\mathbf{r}), \]

where \(\mathbf{u}_{\mathbf{k}}(\mathbf{r})\) is periodic w.r.t. lattice translations \(L_{\mathcal{T}}\). The Bloch function forms an irreps of translation subgroup \(\mathcal{T}\) of space group \(\mathcal{G}\):

\[\begin{split} (\mathbf{E}, \mathbf{t}) \mathbf{\Psi}_{\mathbf{k}}(\mathbf{r}) &= \mathbf{\Psi}( (\mathbf{E}, \mathbf{t})^{-1} \mathbf{r}) \\ &= e^{ -i \mathbf{k} \cdot \mathbf{t} } \mathbf{\Psi}(\mathbf{r}) \quad (\mathbf{t} \in L_{\mathcal{T}}). \end{split}\]

A transformed Bloch function \((\mathbf{R}, \mathbf{v}) \mathbf{\Psi}_{\mathbf{k}}(\mathbf{r})\) is a Bloch function with \(\mathbf{Rk}\):

\[\begin{split} (\mathbf{E}, \mathbf{t}) (\mathbf{R}, \mathbf{v}) \mathbf{\Psi}_{\mathbf{k}}(\mathbf{r}) &= (\mathbf{R}, \mathbf{v}) (\mathbf{E}, \mathbf{R}^{-1}\mathbf{t}) \mathbf{\Psi}_{\mathbf{k}}(\mathbf{r}) \quad (\because z(\mathbf{E}, \mathbf{R}) = z(\mathbf{R}, \mathbf{E}) = 1 ) \\ &= \exp \left( -i \mathbf{k} \cdot \mathbf{R}^{-1} \mathbf{t} \right) (\mathbf{R}, \mathbf{v}) \mathbf{\Psi}_{\mathbf{k}}(\mathbf{r}) \\ &= \exp \left( -i \mathbf{Rk} \cdot \mathbf{t} \right) (\mathbf{R}, \mathbf{v}) \mathbf{\Psi}_{\mathbf{k}}(\mathbf{r}) \quad (\because \mathbf{R} \in \mathrm{SO}(3) ). \end{split}\]

Let \(\Gamma^{\mathbf{k}\alpha}\) be a projective irrep of the little group \(\mathcal{G}^{\mathbf{k}} = \coprod_{ \{ i \mid \mathbf{S}_{i} \in \overline{\mathcal{G}}^{\mathbf{k}} \} } (\mathbf{S}_{i}, \mathbf{w}_{i}) \mathcal{T}\) and \(\{ \mathbf{\Psi}^{\mathbf{k}\alpha}_{\mu} \}_{\mu=1}^{d_{\mathbf{k}\alpha}}\) form the projective irrep. For \(\mathbf{S}_{i} \mathbf{S}_{j} = \mathbf{S}_{k}\),

\[\begin{split} &\sum_{\mu} \mathbf{\Psi}^{\mathbf{k}\alpha}_{\mu} \left[ \mathbf{\Gamma}^{\mathbf{k}\alpha}((\mathbf{S}_{i}, \mathbf{w}_{i})) \mathbf{\Gamma}^{\mathbf{k}\alpha}((\mathbf{S}_{j}, \mathbf{w}_{j})) \right]_{\mu\nu} \\ &= (\mathbf{S}_{i}, \mathbf{w}_{i}) \left( (\mathbf{S}_{j}, \mathbf{w}_{j}) \mathbf{\Psi}^{\mathbf{k}\alpha}_{\nu} \right) \\ &= \frac{ z(\mathbf{S}_{i}, \mathbf{S}_{j}) }{ z(\mathbf{E}, \mathbf{S}_{k}) } (\mathbf{S}_{k}, \mathbf{w}_{k}) \left( (\mathbf{E}, \mathbf{S}_{k}^{-1} (\mathbf{S}_{i}\mathbf{w}_{j} + \mathbf{w}_{i} - \mathbf{w}_{k} ) ) \mathbf{\Psi}^{\mathbf{k}\alpha}_{\nu} \right) \\ &= \sum_{\mu} \mathbf{\Psi}^{\mathbf{k}\alpha}_{\mu} z(\mathbf{S}_{i}, \mathbf{S}_{j}) e^{-i\mathbf{k}\cdot \left( \mathbf{S}_{i}\mathbf{w}_{j} + \mathbf{w}_{i} - \mathbf{w}_{k} \right) } \mathbf{\Gamma}^{\mathbf{k}\alpha}((\mathbf{S}_{k}, \mathbf{w}_{k})) \quad (\because z(\mathbf{E}, \mathbf{S}_{k}) = 1, \mathbf{S}_{k} \in \overline{\mathcal{G}}^{\mathbf{k}}). \end{split}\]

Then, Simplifying by

\[\begin{split} \mathbf{\Gamma}^{\mathbf{k}\alpha}((\mathbf{R}, \mathbf{v})) &=: e^{ -i \mathbf{k} \cdot \mathbf{v} } \mathbf{D}^{\mathbf{k}\alpha}((\mathbf{R}, \mathbf{v})) \\ \mathbf{D}^{\mathbf{k}\alpha}((\mathbf{S}_{i}, \mathbf{w}_{i})) \mathbf{D}^{\mathbf{k}\alpha}((\mathbf{S}_{j}, \mathbf{w}_{j})) &= z(\mathbf{S}_{i}, \mathbf{S}_{j}) e^{ -i \mathbf{g}_{i} \cdot \mathbf{w}_{j} } \mathbf{D}^{\mathbf{k}\alpha}((\mathbf{S}_{k}, \mathbf{w}_{k})), \end{split}\]

where \(\mathbf{g}_{i} = \mathbf{S}_{i}^{-1} \mathbf{k} - \mathbf{k}\). We can enumerate projective irreps for spinor from the factor system \(\mu_{\mathbf{k}}(S_{i}, S_{j}) := z(\mathbf{S}_{i}, \mathbf{S}_{j}) \exp \left( -i \mathbf{g}_{i} \cdot \mathbf{w}_{j} \right)\).

Symmetry operation of the second kind

We assume the inversion \(\mathbf{I}\) acts on spin functions as

\[\begin{split} \mathbf{I} \ket{\uparrow} &= \ket{\uparrow} \\ \mathbf{I} \ket{\downarrow} &= \ket{\downarrow}, \end{split}\]

which is known as Pauli gauge [Alt05]. In this convention, we can choose factor system as follows:

\[\begin{split} z(\mathbf{I}, \mathbf{I}) &= 1 \\ z(\mathbf{R}, \mathbf{I}\mathbf{R}') &= z(\mathbf{IR}, \mathbf{R}') = z(\mathbf{IR}, \mathbf{I}\mathbf{R}') = z(\mathbf{R}, \mathbf{R}') \quad (\mathbf{R}, \mathbf{R}' \in \mathrm{SO}(3)). \end{split}\]

Convention of rotations for spinor

A rotation \(\mathbf{R}_{ \theta\hat{\mathbf{n}} } \in \mathrm{SO}(3)\) can be written with angular momentum operators:

\[\begin{split} \mathbf{R}_{ \theta\hat{\mathbf{n}} } &= \exp \left( -i \theta \hat{\mathbf{n}} \cdot \mathbf{L} \right) \\ &= \begin{pmatrix} \cos \theta + n_{x}^{2} (1 - \cos \theta) & n_{x} n_{y} (1 - \cos \theta) - n_{z} \sin \theta & n_{x} n_{z} (1 - \cos \theta) + n_{y} \sin \theta \\ n_{y} n_{x} (1 - \cos \theta) + n_{z} \sin \theta & \cos \theta + n_{y}^{2} (1 - \cos \theta) & n_{y} n_{z} (1 - \cos \theta) - n_{x} \sin \theta \\ n_{z} n_{x} (1 - \cos \theta) - n_{y} \sin \theta & n_{z} n_{y} (1 - \cos \theta) + n_{x} \sin \theta & \cos \theta + n_{z}^{2} (1 - \cos \theta) \\ \end{pmatrix} \\ \mathbf{L}_{x} &:= -i \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \\ \end{pmatrix} \\ \mathbf{L}_{y} &:= -i \begin{pmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ \end{pmatrix} \\ \mathbf{L}_{z} &:= -i \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} \\ \end{split}\]

The axis and angle of \(\mathbf{R}_{ \theta\hat{\mathbf{n}} }\) are interpret by the following relations:

\[\begin{split} \mathbf{R}_{f} &:= \mathbf{A}^{-1} \mathbf{R}_{ \theta\hat{\mathbf{n}} } \mathbf{A} \quad \in \mathrm{SL}(3) \\ \mathrm{tr} \mathbf{R}_{ \theta\hat{\mathbf{n}} } &= 2 \cos \theta + 1 \\ \end{split}\]

\[\begin{split} \begin{pmatrix} [R_{ \theta\hat{\mathbf{n}} }]_{23} - [R_{ \theta\hat{\mathbf{n}} }]_{32} \\ [R_{ \theta\hat{\mathbf{n}} }]_{31} - [R_{ \theta\hat{\mathbf{n}} }]_{13} \\ [R_{ \theta\hat{\mathbf{n}} }]_{12} - [R_{ \theta\hat{\mathbf{n}} }]_{21} \\ \end{pmatrix} = -2 (\sin \theta) \hat{\mathbf{n}} \end{split}\]

The corresponding unitary rotation is explicitly written as

\[\begin{split} \mathbf{U} ( \mathbf{R}_{ \theta\hat{\mathbf{n}} } ) &= \exp \left( -\frac{i}{2}\theta \hat{\mathbf{n}} \cdot \mathbf{\sigma} \right) \\ &= \mathbf{1} \cos \frac{\theta}{2} - i (\hat{\mathbf{n}} \cdot \mathbf{\sigma}) \sin \frac{\theta}{2} \\ &= \begin{pmatrix} \cos \frac{\theta}{2} - i n_{z} \sin \frac{\theta}{2} & (-i n_{x} - n_{y}) \sin \frac{\theta}{2} \\ (-i n_{x} + n_{y}) \sin \frac{\theta}{2} & \cos \frac{\theta}{2} + i n_{z} \sin \frac{\theta}{2} \\ \end{pmatrix}. \end{split}\]

References

[Alt05]

Simon L Altmann. Rotations, quaternions, and double groups. Courier Corporation, 2005.