Irreps of space group
This page formulates little group, small representation, and irreducible representations (irreps) of space groups.
Irreps of translation subgroup
Consider the following irreps of translation subgroup \(\mathcal{T}\) of space group \(\mathcal{G}\),
\[
\Gamma^{(\mathbf{k})} ((\mathbf{E}, \mathbf{t}))
= \exp \left( -i \mathbf{k} \cdot \mathbf{t} \right)
\quad ( \mathbf{t} \in L_{\mathcal{T}} ),
\]
where \(L_{\mathcal{T}}\) is a lattice formed by \(\mathcal{T}\).
Let \(L_{\mathcal{T}}^{\ast}\) be a reciprocal lattice of \(L_{\mathcal{T}}\).
We can confine \(\mathbf{k}\) within or the surface of a primitive cell of the reciprocal space because, for \(\mathbf{g} \in L_{\mathcal{T}}^{\ast}\), \(\Gamma^{(\mathbf{k})} = \Gamma^{(\mathbf{k} + \mathbf{g})}\).
Any Bloch function with \(\mathbf{k}\)
\[
\Psi_{\mathbf{k}}(\mathbf{r}) = e^{ i \mathbf{k} \cdot \mathbf{r} } u_{\mathbf{k}}(\mathbf{r}),
\]
where \(u_{\mathbf{k}}(\mathbf{r})\) is periodic with \(L_{\mathcal{T}}\), can be taken as (one-dimensional) basis of \(\Gamma^{\mathbf{k}}\):
\[\begin{split}
(\mathbf{E}, \mathbf{t}) \Psi_{\mathbf{k}}(\mathbf{r})
&= \Psi_{\mathbf{k}}( (\mathbf{E}, \mathbf{t})^{-1} \mathbf{r}) \\
&= e^{-i\mathbf{k}\cdot\mathbf{t}} \Psi_{\mathbf{k}}(\mathbf{r}).
\end{split}\]
We write point group of space group \(\mathcal{G}\) as \(\mathcal{P}\).
For \((\mathbf{R}, \mathbf{v}) \in \mathcal{G}\), \((\mathbf{R}, \mathbf{v}) \Psi_{\mathbf{k}}(\mathbf{r})\) is Bloch function with \(\mathbf{Rk}\):
(1)\[\begin{split}
(\mathbf{E}, \mathbf{t}) (\mathbf{R}, \mathbf{v}) \Psi_{\mathbf{k}}(\mathbf{r})
&= (\mathbf{R}, \mathbf{v}) (\mathbf{E}, \mathbf{R}^{-1}\mathbf{t}) \Psi_{\mathbf{k}}(\mathbf{r}) \nonumber \\
&= \exp \left( -i \mathbf{k} \cdot \mathbf{R}^{-1} \mathbf{t} \right) (\mathbf{R}, \mathbf{v}) \Psi_{\mathbf{k}}(\mathbf{r}) \nonumber \\
&= \exp \left( -i \mathbf{Rk} \cdot \mathbf{t} \right) (\mathbf{R}, \mathbf{v}) \Psi_{\mathbf{k}}(\mathbf{r}) \quad (\because \mathbf{R} \in O(3) ).
\end{split}\]
Be careful the basis for calculating inner product in Eq. (1), when you use crystallographic coordinates :
\[\begin{split}
\mathbf{v} &=: \mathbf{A} \mathbf{v}_{f} \\
\mathbf{A}^{-1} \mathbf{R} \mathbf{A} &=: \mathbf{R}_{f} \quad \in \mathrm{SL}(3) \\
(\mathbf{k}, \mathbf{R}^{-1} \mathbf{v})
&= (\mathbf{R} \mathbf{k}, \mathbf{v}) \nonumber \\
&= 2 \pi \left( ( \mathbf{A}^{\top} \mathbf{A}^{-1} ) \mathbf{R}_{f} ( \mathbf{A}^{\top} \mathbf{A}^{-1} )^{-1} \mathbf{k}_{f}, \mathbf{v}_{f} \right) \nonumber \\
&= 2 \pi \left( \mathbf{R}_{f}^{\top} \mathbf{k}_{f}, \mathbf{v}_{f} \right)
\quad (\because \mathbf{R}_{f}^{\top} = \mathbf{A}^{\top} \mathbf{R}^{-1} \mathbf{A}^{-\top} ).
\end{split}\]
Little group
Let \(\mathcal{P}\) be point group of space group \(\mathcal{G}\).
We define little co-group of \(\mathbf{k}\),
\[
\overline{\mathcal{G}}^{\mathbf{k}} = \left\{ \mathbf{R} \in \mathcal{P} \mid \mathbf{Rk} \equiv \mathbf{k} \right\},
\]
where \(\equiv\) is up to reciprocal vectors in \(L_{\mathcal{T}}^{\ast}\).
When we use crystallographic coordinates w.r.t. primitive basis vectors, the condition \(\mathbf{Rk} \equiv \mathbf{k}\) is written as
\[
\mathbf{Rk} \equiv \mathbf{k}
\Longleftrightarrow
\mathbf{R}_{f}^{\top} \mathbf{k}_{f} - \mathbf{k}_{f} \in \mathbb{Z}^{3}.
\]
Because lattice \(L_{\mathcal{T}}\) is invariant with \(\mathbf{R} \in \mathcal{P}\), we obtain
\[
\mathbf{R} \mathbf{k}_{1} \equiv \mathbf{k}_{2}
\Rightarrow
\overline{\mathcal{G}}^{\mathbf{k}_{2}} = \mathbf{R} \overline{\mathcal{G}}^{\mathbf{k}_{1}} \mathbf{R}^{-1}.
\]
When we decompose point group \(\mathcal{P}\) as
\[
\mathcal{P} = \coprod_{i=1}^{q} \mathbf{P}_{i} \overline{\mathcal{G}}^{\mathbf{k}_{1}},
\]
\(\mathbf{P}_{i}\mathbf{k}_{1} \equiv \mathbf{k}_{i}\) with star of \(\mathbf{k}_{1}\), \(\left\{ \mathbf{k}_{1}, \dots, \mathbf{k}_{q} \right\}\).
We define little group (of the first kind) of \(\mathbf{k}\) as
\[
\mathcal{G}^{\mathbf{k}} = \coprod_{ \{ i \mid \mathbf{S}_{i} \in \overline{\mathcal{G}}^{\mathbf{k}} \} } (\mathbf{S}_{i}, \mathbf{w}_{i}) \mathcal{T},
\]
where \(\mathcal{G} = \coprod_{i} (\mathbf{S}_{i}, \mathbf{w}_{i}) \mathcal{T}\).
We can decompose space group \(\mathcal{G}\) as
\[
\mathcal{G} = \coprod_{i=1}^{q} (\mathbf{P}_{i}, \mathbf{x}_{i}) \mathcal{G}^{\mathbf{k}_{1}}.
\]
Then, for Bloch function \(\Psi_{\mathbf{k}_{1}}(\mathbf{r})\), \((\mathbf{P}_{i}, \mathbf{x}_{i}) \mathcal{G}^{\mathbf{k}_{1}} \Psi_{\mathbf{k}_{1}}(\mathbf{r})\) is Bloch function for \(\mathbf{k}_{i}\).
Thus, our task to calculate irreps of \(\mathcal{G}\) is boiled down to calculating irreps of little group \(\mathcal{G}^{\mathbf{k}}\), which is called small representation.
Projective representation
A projective representation of group \(G\) is a non-singular matrix function \(\Delta\) on \(G\) if it satisfied the following conditions:
For each group product \(G_{k} = G_{i}G_{j}\), there exists scalar function \(\mu\) (factor system) such that
\[
\Delta(G_{i}) \Delta(G_{j}) = \mu(G_{i}, G_{j}) \Delta(G_{k}),
\]
and
\[
\mu(G_{i}, G_{j}G_{k}) \mu(G_{j}, G_{k}) = \mu(G_{i}G_{j}, G_{k}) \mu(G_{i}, G_{j})
\quad \mbox{(cocycle condition)}.
\]
Small representation
Let \(\Gamma^{\mathbf{k}}_{p}\) be irrep of \(\mathcal{G}^{\mathbf{k}}\), for \(\mathbf{S}_{i} \mathbf{S}_{j} = \mathbf{S}_{k}\),
\[
\mathbf{\Gamma}^{\mathbf{k}}_{p}((\mathbf{S}_{i}, \mathbf{w}_{i})) \mathbf{\Gamma}^{\mathbf{k}}_{p}((\mathbf{S}_{j}, \mathbf{w}_{j}))
= \exp \left( -i \mathbf{k} \cdot ( \mathbf{w}_{i} + \mathbf{S}_{i} \mathbf{w}_{j} - \mathbf{w}_{k} ) \right) \mathbf{\Gamma}^{\mathbf{k}}_{p}((\mathbf{S}_{k}, \mathbf{w}_{k})).
\]
Then, Simplifying by
\[\begin{split}
\mathbf{\Gamma}^{\mathbf{k}}_{p}((\mathbf{R}, \mathbf{v})) &=: \exp \left( -i \mathbf{k} \cdot \mathbf{v} \right) \mathbf{D}^{\mathbf{k}}_{p}((\mathbf{R}, \mathbf{v})) \\
\mathbf{D}^{\mathbf{k}}_{p}((\mathbf{S}_{i}, \mathbf{w}_{i})) \mathbf{D}^{\mathbf{k}}_{p}((\mathbf{S}_{j}, \mathbf{w}_{j}))
&= \exp \left( -i \mathbf{g}_{i} \cdot \mathbf{w}_{j} \right) \mathbf{D}^{\mathbf{k}}_{p}((\mathbf{S}_{k}, \mathbf{w}_{k})),
\end{split}\]
where
\[\begin{split}
\mathbf{g}_{i}
&= \mathbf{S}_{i}^{-1} \mathbf{k} - \mathbf{k} \\
&= 2 \pi \mathbf{B} \left( \mathbf{S}_{i, f}^{\top} \mathbf{k}_{f} - \mathbf{k}_{f} \right)
\quad (\mathbf{S}_{i, f} = \mathbf{A} \mathbf{S}_{i} \mathbf{A}^{-1}).
\end{split}\]
If we choose origin such that \((\mathbf{E}, \mathbf{t}) \in \mathcal{G}\), \(\mathbf{D}^{\mathbf{k}}_{p}((\mathbf{E}, \mathbf{t})) = \mathbf{1}\) and \(\exp \left( -i \mathbf{g}_{i} \cdot (\mathbf{w}_{j} + \mathbf{t}) \right) =\exp \left( -i \mathbf{g}_{i} \cdot \mathbf{w}_{j} \right) \) hold.
Then, it is sufficient to consider on \(\overline{\mathcal{G}}^{\mathbf{k}}\),
\[
\mathbf{D}^{\mathbf{k}}_{p}(\mathbf{S}_{i}) \mathbf{D}^{\mathbf{k}}_{p}(\mathbf{S}_{j})
= \exp \left( -i \mathbf{g}_{i} \cdot \mathbf{w}_{j} \right) \mathbf{D}^{\mathbf{k}}_{p}(\mathbf{S}_{k})
\]
Here, \(\mu_{\mathbf{k}}(S_{i}, S_{j}) := \exp \left( -i \mathbf{g}_{i} \cdot \mathbf{w}_{j} \right)\) holds the cocycle condition and \(\mu_{\mathbf{k}}(\mathbf{E}, \mathbf{E}) = 1\).
Now, our task is to enumerate projective irreps of little co-group \(\overline{\mathcal{G}}^{\mathbf{k}}\) with factor system \(\mu_{\mathbf{k}}\).
When we use crystallographic coordinates, the factor system is written as
\[\begin{split}
\mu_{\mathbf{k}}(\mathbf{S}_{i}, \mathbf{S}_{j})
&= \exp \left( -i (\mathbf{g}_{i}, \mathbf{w}_{j}) \right) \\
&= \exp \left( -2 \pi i (\mathbf{S}_{i, f}^{\top} \mathbf{k}_{f} - \mathbf{k}_{f}, \mathbf{w}_{j, f}) \right)
\quad (\mathbf{w}_{j} =: \mathbf{A} \mathbf{w}_{j, f}).
\end{split}\]