Supercell construction#

Klassengleiche subgroup#

Let \(L(\mathcal{G})\) be a lattice of space group \(\mathcal{G}\),

\[ L(\mathcal{G}) = \left\{ \mathbf{t} \mid (\mathbf{I}, \mathbf{t}) \in \mathcal{T}(\mathcal{G}) \right\}.\]

Let \(\mathcal{P} := \mathcal{P}(\mathcal{G})\) be a point group of space group \(\mathcal{G}\). Lattice \(L (\subset L(\mathcal{G}))\) is called \(\mathcal{P}\)-invariant if \(\mathcal{P}\) acts faithfully on \(L\) with the following action, \(\mathcal{P} \times L \ni (\mathbf{R}, \mathbf{t}) \mapsto \mathbf{Rt} \in L\).

Theorem 5.2 in Ref. [EGahlerN97]

Let \(\mathcal{M}\) be a maximal k-subgroup of space group \(\mathcal{G}\):

  1. \(\mathcal{T}(\mathcal{M})\) is a maximal \(\mathcal{G}\)-invariant subgroup of \(\mathcal{T}(\mathcal{G})\).

  2. \(\mathcal{T}(\mathcal{G}) / \mathcal{T}(\mathcal{M})\) is an elementary abelian \(p\)-group for a prime \(p\). That is, \(\mathcal{T}(\mathcal{G}) / \mathcal{T}(\mathcal{M}) \simeq (\mathbb{Z} / p \mathbb{Z})^{r}\) for a non-negative integer \(r\).

The above theorem can be proved as follows. When \(\mathcal{M}\) is a maximal k-subgroup, \(\mathcal{T}(\mathcal{G}) / \mathcal{T}(\mathcal{M})\) has no proper subgroup that is invariant under conjugate action of \(\mathcal{G}/\mathcal{T}(\mathcal{M})\). Thus, \(\mathcal{T}(\mathcal{G}) / \mathcal{T}(\mathcal{M})\) has no charactristic subgroup 1. In general, finitely generated abelian group without charastristic subgroups is an elementary abelian \(p\)-group for a prime \(p\) 2.

We need to find a translation subgroup \(\mathcal{S} < \mathcal{T}(\mathcal{G})\) with

  • \(\mathcal{T}(\mathcal{G}) / \mathcal{S} \simeq \mathbb{Z}_{p^{r}}\) for some prime number \(p\) and integer \(r\).

  • Let \(L(\mathcal{S}) = \mathbf{M}\mathbb{Z}^{n}\). For all \(\mathbf{R} \in \mathcal{P}(\mathcal{G})\), \(\mathbf{RM}\mathbb{Z}^{n} = \mathbf{M}\mathbb{Z}^{n}\), that is, \(\mathbf{M}^{-1}\mathbf{R}\mathbf{M}\) is a unimodular matrix.

Let the Smith normal form of \(\mathbf{M}\) as \(\mathbf{M} = \mathrm{diag}(n_{1}, n_{2}, n_{3})\). The condition that \(\mathcal{T}(\mathcal{G}) / \mathcal{S}\) has no proper subgroups requires \(n_{1} = n_{2} = 1\) and \(n_{3}\) is a power of a prime number, which follow from the Chinese remainder theorem.

Minimal supercell problem#

Reference [FKCM19] proposed a method to find supercell matrix \(\mathbf{H}\) that accommodates all given wave vectors \(\mathbf{Q}\):

\[ \min_{ \mathbf{H} \in \mathbb{Z}^{d \times d} } \det \mathbf{H} \quad \mathrm{s.t.} \quad \mathbf{H}^{\top} \mathbf{q} \equiv \mathbf{0}_{d} \, (\mathrm{mod}\, \mathbb{Z}^{d}) \quad (\forall \mathbf{q} \in \mathbf{Q})\]

Ref. [LWM15] gave a specific formula for a commensurate supercell for given single wave vector.

References#

EGahlerN97

B. Eick, F. Gähler, and W. Nickel. Computing Maximal Subgroups and Wyckoff Positions of Space Groups. Acta Crystallographica Section A, 53(4):467–474, Jul 1997. URL: https://doi.org/10.1107/S0108767397003462, doi:10.1107/S0108767397003462.

FKCM19

Lyuwen Fu, Mordechai Kornbluth, Zhengqian Cheng, and Chris A. Marianetti. Group theoretical approach to computing phonons and their interactions. Phys. Rev. B, 100:014303, Jul 2019. URL: https://link.aps.org/doi/10.1103/PhysRevB.100.014303, doi:10.1103/PhysRevB.100.014303.

LWM15

Jonathan H. Lloyd-Williams and Bartomeu Monserrat. Lattice dynamics and electron-phonon coupling calculations using nondiagonal supercells. Phys. Rev. B, 92:184301, Nov 2015. URL: https://link.aps.org/doi/10.1103/PhysRevB.92.184301, doi:10.1103/PhysRevB.92.184301.

1

A charastiristic subgroup of group \(G\) is a subgroup that is invariant under all authmorphism of \(G\).

2

If \(\mathrm{Aut}(G)\) acts on \(G\) transitively, all elements other than identity have the same prime order.