Contents

Isotropy subgroup

Consider space group \(\mathcal{G}\) and its representation

\[ g \phi_{j} = \sum_{i} \phi_{i} \Gamma(g)_{ij} \quad (g \in \mathcal{G}).\]

A subgroup \(\mathcal{H} (\leq \mathcal{G})\) is called isotropy subgroup belonging \(\Gamma\) if the subduced representation \(\Gamma \downarrow \mathcal{H}\) has an identity representation 1 [HS98, SH89, SHW91].

When we consider a linear combination of basis functions as \(\mathbf{\eta} = \sum_{i} \eta_{i} \phi_{i}\), we call \(\mathbf{\eta} = \{ \eta_{i} \}_{i}\) as order parameters. Two order parameters \(\mathbf{\eta}\) and \(\mathbf{\eta}'\) are equivalent if some operation \(g \in \mathcal{G}\) exists such that

\[ \mathbf{\eta}' = \mathbf{\Gamma}(g) \mathbf{\eta}.\]

Algorithm to generate isotropy subgroups of space group [SC17]

For a small representation \(\Gamma^{\mathbf{k}\alpha}\) of space group \(\mathcal{G}\), its maximal isotropy subgroups \(\mathcal{S}\) are enumerated as follows.

Determine sublattice \(\mathcal{L}_{\mathcal{S}}\) and translational subgroup \(\mathcal{T}(\mathcal{S})\)

The requirements that \(\mathcal{S}\) is an isotropy subgroup of \(\mathcal{G}\) is

\[ \exists \mathbf{\eta} \neq \mathbf{0} \, s.t. \, \mathbf{\Gamma}^{\mathbf{k}\alpha}(g) \mathbf{\eta} = \mathbf{\eta} \quad (\forall g \in \mathcal{S}).\]

In particular, if translation \((\mathbf{E}, \mathbf{t})\) belongs to \(\mathcal{S}\), \(e^{-i\mathbf{k}\cdot\mathbf{t}} = 1\). Thus, a translational subgroup of \(\mathcal{S}\) should be

\[ \mathcal{T}(\mathcal{S}) := \{ (\mathbf{E}, \mathbf{t}) \in \mathcal{G} \mid \mathbf{k} \cdot \mathbf{t} \in 2\pi \mathbb{Z} \}.\]

For later use, let \(\mathcal{L}_{\mathcal{S}}\) be a sublattice \(\mathcal{L}_{\mathcal{S}}\) formed by translation parts in \(\mathcal{T}(\mathcal{S})\). Note that, although subgroups of \(\mathcal{T}(\mathcal{S})\) also satisfy the requirements, there is no need to consider such subgroups because these subgroups show at lower-symmetry \(\mathbf{k}\) vector.

There are two basis vectors for \(\mathcal{T}(\mathcal{S})\) that are orthogonal to \(\mathbf{k}\). Let \(l\) be LCM of denominators of \(\frac{1}{2\pi}\mathbf{k}\). Then, we write the elements of \(\mathbf{k}\) as

\[ \mathbf{k} = 2\pi \left( \frac{g a_{1}}{l} \frac{g a_{2}}{l} \frac{g a_{3}}{l} \right)^{\top},\]

where \(\mathrm{GCD}(a_{1}, a_{2}, a_{3}) = 1\), \(\mathrm{GCD}(g, l) = 1\) and \(1 \leq g < l\).

\[\begin{split} &\mathbf{k} \cdot \mathbf{t} \in 2\pi \mathbb{Z} \\ &\Leftrightarrow g (a_{1} \,a_{2} \,a_{3}) \mathbf{t} \equiv 0 \quad (\mathrm{mod}\, l) \\ &\Leftrightarrow (a_{1} \,a_{2} \,a_{3}) \mathbf{t} \equiv 0 \quad (\mathrm{mod}\, l) \quad (\because \mathrm{GCD}(g, l) = 1) \\\end{split}\]

By solving \((a_{1} \,a_{2} \,a_{3}) \mathbf{t} = l\), we obtain one special solution \((a_{1} \,a_{2} \,a_{3}) \mathbf{t}_{0} = l\) and two general solutions \((a_{1} \,a_{2} \,a_{3}) \mathbf{t}_{i} = 0 \, (i=1,2)\). Then, \(\{ n\mathbf{t}_{0}, \mathbf{t}_{1}, \mathbf{t}_{2} \}\) spans a lattice

\[ \{ \mathbf{t} \in \mathbb{Z}^{3} \mid (a_{1} \,a_{2} \,a_{3}) \mathbf{t} = nl \}\]

for \(n \in \mathbb{Z}\). Thus, \(\{ \mathbf{t}_{0}, \mathbf{t}_{1}, \mathbf{t}_{2} \}\) is basis of \(\mathcal{L}_{S}\). By stacking these basis vectors, we can find a transformation matrix \(\mathbf{M}\),

\[ \mathcal{L}_{\mathcal{S}} = \{ \mathbf{Mt} \mid (\mathbf{E}, \mathbf{t}) \in \mathcal{G} \}.\]

Enumerate point group \(\mathcal{P}(\mathcal{S})\)

Next, we find a subgroup of \(\mathcal{P}(\mathcal{G})\) which preserve the sublattice \(\mathcal{L}_{\mathcal{S}}\),

\[\begin{split} \mathcal{B}(\mathcal{S}) &:= \{ \mathbf{R} \in \mathcal{P}(\mathcal{G}) \mid \mathbf{R} \mathcal{L}_{\mathcal{S}} =\mathcal{L}_{\mathcal{S}} \} \\ &= \{ \mathbf{R} \in \mathcal{P}(\mathcal{G}) \mid \mathbf{M}^{-1}\mathbf{R}\mathbf{M} \,\mbox{is unimodular} \} \\\end{split}\]

Point group of isotropy subgroup \(\mathcal{S}\) should be a subgroup of \(\mathcal{B}(\mathcal{S})\).

Enumerate isotropy subgroup \(\mathcal{S}\)

For given point group \(\mathcal{P}(\mathcal{S})\) and translational subgroup \(\mathcal{T}(\mathcal{S})\), consider the following set

\[ \mathcal{S} := \{ ( \mathbf{R}_{i}, \mathbf{\tau}_{i} + \mathbf{c}_{i} + \mathbf{l} ) \mid \mathbf{R}_{i} \in \mathcal{P}(\mathcal{S}), \mathbf{l} \in \mathcal{L}_{\mathcal{S}} \}.\]

Here \(\mathbf{c}_{i} \in \mathcal{T}(\mathcal{G})\) can be freely chosen.

The condition that \(\mathcal{S}\) is a subgroup of \(\mathcal{G}\) is as follows:

\[\begin{split} &\forall ( \mathbf{R}_{i}, \mathbf{\tau}_{i} + \mathbf{Mt} ), ( \mathbf{R}_{j}, \mathbf{\tau}_{j} + \mathbf{Mt}' ) \in \mathcal{S}, ( \mathbf{R}_{i}, \mathbf{\tau}_{i} + \mathbf{Mt} )^{-1} ( \mathbf{R}_{j}, \mathbf{\tau}_{j} + \mathbf{Mt}' ) \in \mathcal{S} \\ &\Leftrightarrow \forall ( \mathbf{R}_{i}, \mathbf{\tau}_{i} + \mathbf{Mt} ), ( \mathbf{R}_{j}, \mathbf{\tau}_{j} + \mathbf{Mt}' ) \in \mathcal{S}, \mathbf{\tau}_{j} + \mathbf{Mt}' - \mathbf{R}_{i}^{-1}(\mathbf{\tau}_{i} + \mathbf{Mt}) \in \mathcal{L}_{\mathcal{S}} \\ &\Leftrightarrow \forall \mathbf{R}_{i}, \mathbf{R}_{j} \in \mathcal{S}, \exists k \,s.t.\, \mathbf{R}_{i}^{-1} \mathbf{R}_{j} = \mathbf{R}_{k}, \mathbf{\tau}_{j} - \mathbf{R}_{i}^{-1}\mathbf{\tau}_{i} - \mathbf{\tau}_{k} \in \mathcal{L}_{\mathcal{S}} \\\end{split}\]

Determine order-parameter direction

Non-zero order-parameter directions correspond to eigenvectors with eigenvalue 1 of the projection operator,

\[ \frac{1}{|\overline{\mathcal{S}}|} \sum_{ g \in \overline{\mathcal{S}} } \Gamma^{\mathbf{k}\alpha}(g).\]

References

HS98

C. J. Howard and H. T. Stokes. Group-Theoretical Analysis of Octahedral Tilting in Perovskites. Acta Crystallographica Section B, 54(6):782–789, Dec 1998. URL: https://doi.org/10.1107/S0108768198004200, doi:10.1107/S0108768198004200.

SH89

H T Stokes and D M Hatch. Isotropy Subgroups of the 230 Crystallographic Space Groups. WORLD SCIENTIFIC, edition, 1989. URL: https://www.worldscientific.com/doi/abs/10.1142/0751, doi:10.1142/0751.

SC17

Harold T. Stokes and Branton J. Campbell. A general algorithm for generating isotropy subgroups in superspace. Acta Crystallographica Section A, 73(1):4–13, Jan 2017. URL: https://doi.org/10.1107/S2053273316017629, doi:10.1107/S2053273316017629.

SHW91

Harold T. Stokes, Dorian M. Hatch, and James D. Wells. Group-theoretical methods for obtaining distortions in crystals: applications to vibrational modes and phase transitions. Phys. Rev. B, 43:11010–11018, May 1991. URL: https://link.aps.org/doi/10.1103/PhysRevB.43.11010, doi:10.1103/PhysRevB.43.11010.

1

Consider group \(G\) acting on space \(X\). In general, isotropy subgroup is defined as stabilizer of point \(x\) in \(X\) as

\[ G_{x} = \left\{ g \in G \mid g x = x \right\}.\]
Lattice vibration API Reference