Crystal symmetry & space groups

The group theory that band and phonon theory rest on. A crystal Hamiltonian commutes with a discrete group of rigid motions — the space group $\mathcal{G}$ — so its eigenstates carry irreducible representations (irreps) of $\mathcal{G}$, and every degeneracy that is not an accident of parameters is the dimension of one of those irreps. The whole apparatus below answers two operational questions I keep returning to: which band or phonon contacts are forced by symmetry, and how few independent quantities (force constants, hopping integrals, atomic displacements) a calculation actually has to determine. The first is read off from the irreps of the little group of a wavevector; the second from the orbit structure of the symmetry action on atoms and on the force-constant tensor. This is why the dynamical matrix has exactly the degeneracies it does at high-symmetry points, and why a finite-displacement code displaces only symmetry-inequivalent atoms in symmetry-inequivalent directions.

Space groups and their action

A space-group element is a Seitz operator $\{R|\mathbf{t}\}$ acting on real space as

$$\{R|\mathbf{t}\}\mathbf{r}=R\mathbf{r}+\mathbf{t},\{R|\mathbf{t}\}\{R^{\prime }|{\mathbf{t}}^{\prime }\}=\{R{R}^{\prime }|R{\mathbf{t}}^{\prime }+\mathbf{t}\},$$

with $R$ a point operation (a member of the crystallographic point group, $\det R=\pm 1$) and $\mathbf{t}$ a translation. The pure lattice translations $\{E|\mathbf{R}\}$, $\mathbf{R}$ a Bravais vector, form an invariant abelian subgroup $\mathcal{T}◃\mathcal{G}$, and the quotient

$$\mathcal{G}/\mathcal{T}\cong \mathcal{P}$$

is the point group $\mathcal{P}$ of order $|\mathcal{P}|=h$ — the $R$‘s that appear, stripped of their translations. A space group is symmorphic if it has a setting in which every $\mathbf{t}$ is a lattice vector, so $\mathcal{G}=\mathcal{T}⋊\mathcal{P}$ is a genuine semidirect product; otherwise it is non-symmorphic and some operations carry an unavoidable fractional translation $\mathbf{t}={\mathbf{v}}_R+\mathbf{R}$ with ${\mathbf{v}}_R\notin \mathcal{T}$ — a screw axis (rotation

• fractional translation along it) or a glide plane (mirror + fractional translation in it). These fractional translations are not cosmetic: they are what protect the stick-together band and phonon degeneracies at Brillouin-zone boundaries (the nonsymmorphic “extra” contacts), and they are the reason the little-group representations at the zone edge can be projective rather than ordinary. Of the 230 three-dimensional space groups, 73 are symmorphic.

Bloch states as a representation of $\mathcal{T}$

Because $\mathcal{T}$ is abelian, its irreps are one-dimensional and labelled by a crystal momentum $\mathbf{k}$ in the first Brillouin zone (BZ):

$$\{E|\mathbf{R}\}:|\psi ⟩\mapsto e^{-i\mathbf{k}\cdot \mathbf{R}}|\psi ⟩.$$

This is exactly Bloch’s theorem read group-theoretically — $\mathbf{k}$ is the label of the irrep of the translation subgroup that a state transforms in. The full space group then acts on this label. For $g=\{R|\mathbf{t}\}\in \mathcal{G}$ and a Bloch state $|{\psi }_{\mathbf{k}}⟩$, a one-line computation using the multiplication rule shows

$$\hat{g}|{\psi }_{\mathbf{k}}⟩\text{transforms under }\mathcal{T}\text{ with momentum }R\mathbf{k}:\hat{g}:\mathbf{k}\mapsto R\mathbf{k},$$

i.e. the point part $R$ permutes the Bloch sectors $\mathbf{k}\to R\mathbf{k}$ (mod a reciprocal lattice vector $\mathbf{G}$, since $\mathbf{k}$ lives in the BZ). The space group therefore does not act within a single $\mathbf{k}$; it shuffles a whole set of $\mathbf{k}$‘s into one another, and only a subgroup fixes a given $\mathbf{k}$.

The little group and the star of $\mathbf{k}$

The little group (little co-group, group of the wavevector) is the stabilizer of $\mathbf{k}$:

$${\mathcal{G}}_{\mathbf{k}}=\{\{R|\mathbf{t}\}\in \mathcal{G}:R\mathbf{k}=\mathbf{k}+\mathbf{G}\},$$

the operations that map $\mathbf{k}$ to itself modulo a reciprocal-lattice vector. Its point part ${\mathcal{P}}_{\mathbf{k}}\subseteq \mathcal{P}$ is the little point group. The orbit of $\mathbf{k}$ under the full point group,

$${}^{\ast }\mathbf{k}=\{R\mathbf{k}:R\in \mathcal{P}\},$$

is the star of $\mathbf{k}$; by orbit–stabilizer its size is the index

$$|{}^{\ast }\mathbf{k}|=[\mathcal{P}:{\mathcal{P}}_{\mathbf{k}}]=\frac{h}{|{\mathcal{P}}_{\mathbf{k}}|}.$$

At a generic (low-symmetry) $\mathbf{k}$ the little group is just $\mathcal{T}$ (plus, whenever some spatial operation $R$ — inversion, a two-fold axis, or a mirror — sends $\mathbf{k}\to -\mathbf{k}$, the corresponding antiunitary element $\Theta R$ supplied by time reversal), the star has the full $h$ arms, and the bands are non-degenerate. At a high-symmetry point $\mathbf{k}$ — $\Gamma$, a zone-boundary point, or a point on a symmetry line — ${\mathcal{P}}_{\mathbf{k}}$ grows and the star shrinks. The physical irreps of the full space group at momentum $\mathbf{k}$ are built by induction from the irreps of the little group: an irrep of $\mathcal{G}$ associated with the star ${}^{\ast }\mathbf{k}$ is ${\mathrm{Ind}}_{{\mathcal{G}}_{\mathbf{k}}}^{\mathcal{G}}{D}_{\mathbf{k}}$, of dimension $|{}^{\ast }\mathbf{k}|\times \dim D_{\mathbf{k}}$ (one copy of $D_{\mathbf{k}}$ on each arm of the star). For working purposes all the labelling and degeneracy information at a single $\mathbf{k}$ therefore lives in the small (allowed) irreps $D_{\mathbf{k}}$ of ${\mathcal{G}}_{\mathbf{k}}$.

A subtlety that matters at zone boundaries: for a non-symmorphic group the little-group elements carry their fractional translations $\{R|{\mathbf{v}}_R\}$, and the relevant representations are the $\mathbf{k}$-dependent ones obeying

$$D_{\mathbf{k}}(\{R|{\mathbf{v}}_R\})D_{\mathbf{k}}(\{R^{\prime }|{\mathbf{v}}_{R^{\prime }}\})=e^{-i(R\mathbf{k}-\mathbf{k})\cdot {\mathbf{v}}_{R^{\prime }}}{D}_{\mathbf{k}}(\{R{R}^{\prime }|\dots \}),$$

a projective (ray) representation whenever the phase factor is nontrivial — generic on the BZ surface. These extra phases enlarge the minimal dimension of the allowed irreps and so force band/phonon sticking at the zone edge; this is the group-theoretic origin of the nonsymmorphic degeneracies. (Below I keep $\mathbf{k}$ in the interior, where the phases are trivial and ${\mathcal{P}}_{\mathbf{k}}$ acts like an ordinary point group, except where stated.)

Compatibility relations

The little group changes discontinuously as $\mathbf{k}$ moves: it is large at a high-symmetry point and smaller on the lines and planes radiating from it, with the smaller groups being subgroups of the larger, ${\mathcal{P}}_{{\mathbf{k}}_{\text{line}}}\subset {\mathcal{P}}_{{\mathbf{k}}_{\text{point}}}$. Compatibility relations are the subduction (restriction) of each point’s irreps onto the line’s:

$$D_{{\mathbf{k}}_{\text{point}}}^{(\mu )}{\downarrow }_{{\mathcal{P}}_{{\mathbf{k}}_{\text{line}}}}=\underset{\nu }{⨁}{m}_{\mu \nu }{D}_{{\mathbf{k}}_{\text{line}}}^{(\nu )},m_{\mu \nu }=\frac{1}{|{\mathcal{P}}_{{\mathbf{k}}_{\text{line}}}|}\sum _{g\in {\mathcal{P}}_{{\mathbf{k}}_{\text{line}}}}{\chi }^{(\mu )}(g{)}^{\ast }{\chi }^{(\nu )}(g),$$

the standard reduction formula with the characters $\chi$ of the two groups. Physically: a level of symmetry $\mu$ at the high-symmetry point must split into precisely the line irreps $\nu$ appearing in this branching as you step away, and conversely two line branches can only reconnect at a point into an irrep that subduces to both. Tracking these relations along a $\mathbf{k}$-path is what lets you connect a band/phonon structure into continuous, non-crossing (within an irrep) sheets and decide whether a crossing on a line is symmetry-allowed (different $\nu$) or must gap (same $\nu$). It is the bookkeeping behind every correctly connected band diagram, and the first sieve for whether a contact is robust — the entry point to symmetry-protected band/phonon topology, which the tight-binding and chiral-phonon analyses pick up.

Site symmetry, Wyckoff positions, and the equivalent-atom orbit

The same group acts on the atoms. The site-symmetry group ${\mathcal{S}}_{\mathbf{x}}$ of a point $\mathbf{x}$ in the cell is the subgroup of $\mathcal{G}$ that fixes $\mathbf{x}$,

$${\mathcal{S}}_{\mathbf{x}}=\{g\in \mathcal{G}:g\mathbf{x}=\mathbf{x}\},$$

necessarily isomorphic to a crystallographic point group (it can contain no nontrivial pure translation). The orbit $\mathcal{G}\mathbf{x}$ of symmetry-equivalent points is a Wyckoff position; by orbit–stabilizer its multiplicity per primitive cell is

$$m_{\mathbf{x}}=\frac{|\mathcal{P}|}{|{\mathcal{S}}_{\mathbf{x}}|}.$$

A special position sits on one or more symmetry elements, so ${\mathcal{S}}_{\mathbf{x}}$ is nontrivial and $m_{\mathbf{x}}$ is reduced; a general position has ${\mathcal{S}}_{\mathbf{x}}=\{E\}$ and full multiplicity $|\mathcal{P}|$. Two consequences I lean on constantly:

• Only inequivalent atoms (and inequivalent displacement directions) carry independent data. The force constants, hopping integrals, multipoles, etc. on the atoms of one Wyckoff orbit are related by the space-group operations to those on a single representative. A finite-displacement phonon calculation therefore needs to perturb only one atom per orbit, and only along the directions not related by its site symmetry ${\mathcal{S}}_{\mathbf{x}}$: the remaining columns of the force-constant matrix are generated by symmetry. This is exactly the optimization finite-displacement phonons (phonopy and kin) perform — the number of independent supercell calculations is set by orbits and site symmetry, not by the raw atom count.
• Site symmetry seeds orbital/band content. The local representation furnished by an atom’s orbitals at $\mathbf{x}$ is a representation of ${\mathcal{S}}_{\mathbf{x}}$; inducing it up the chain ${\mathcal{S}}_{\mathbf{x}}\to \mathcal{G}\to {\mathcal{G}}_{\mathbf{k}}$ gives the band representation — the symmetry content the orbitals are required to produce throughout the BZ. (Bands that cannot be written as such an induced representation are the topological/obstructed ones; this is the topological-quantum-chemistry viewpoint, and it is downstream of the little-group machinery here.)

Symmetry-adapted normal modes at a high-symmetry point

The cleanest payoff is the symmetry-adapted (normal-mode) decomposition, which says in advance how a phonon or tight-binding block at a high-symmetry $\mathbf{k}$ splits, and supplies a basis that makes it block-diagonal. Take the $3N$-dimensional space of atomic displacements in the cell (or the $M$-dimensional space of orbitals). The little point group ${\mathcal{P}}_{\mathbf{k}}$ acts on it through the mechanical (displacement) representation ${\Gamma }_{\mathbf{k}}^{\text{mech}}$, whose character is computed from a single geometric rule — only atoms left in place by $g$ (up to a lattice translation, with the appropriate Bloch phase at $\mathbf{k}\ne \Gamma$) contribute, weighted by how a polar vector transforms under $g$:

$${\chi }_{\mathbf{k}}^{\text{mech}}(g)=\sum _{a:g{\mathbf{x}}_a={\mathbf{x}}_a(\mathrm{mod}\mathcal{T})}{e}^{i\mathbf{k}\cdot \Delta {\mathbf{R}}_a(g)}[\pm 1+2\cos {\theta }_g],$$

with ${\theta }_g$ the rotation angle of $R$ and the sign $+$ for proper, $-$ for improper operations ($[\pm 1+2\cos \theta ]$ is the character of the vector representation). Reducing it,

$${\Gamma }_{\mathbf{k}}^{\text{mech}}=\underset{\mu }{⨁}{n}_{\mu }{D}_{\mathbf{k}}^{(\mu )},n_{\mu }=\frac{1}{|{\mathcal{P}}_{\mathbf{k}}|}\sum _{g\in {\mathcal{P}}_{\mathbf{k}}}{\chi }_{\mathbf{k}}^{(\mu )}(g{)}^{\ast }{\chi }_{\mathbf{k}}^{\text{mech}}(g),$$

tells you immediately the irrep labels of the modes and their degeneracies: each $D_{\mathbf{k}}^{(\mu )}$ of dimension $d_{\mu }$ contributes an $n_{\mu }$-fold set of $d_{\mu }$-fold degenerate modes. A two- or three-dimensional irrep is a symmetry-enforced double or triple degeneracy — this is precisely the source of the degeneracies the dynamical matrix exhibits at $\Gamma$ and at the high-symmetry zone points; the dynamical matrix, restricted to such a $\mathbf{k}$, must commute with every $D_{\mathbf{k}}(g)$, so by Schur’s lemma it is constant ($\propto 1$) on each irrep block and its eigenvalues come in the multiplicities $d_{\mu }$. The explicit projection operators

$${\hat{P}}_{rs}^{(\mu )}=\frac{d_{\mu }}{|{\mathcal{P}}_{\mathbf{k}}|}\sum _{g\in {\mathcal{P}}_{\mathbf{k}}}{D}_{\mathbf{k}}^{(\mu )}(g{)}_{rs}^{\ast }\hat{g}$$

applied to the displacement (or orbital) basis return the symmetry-adapted vectors that bring $D(\mathbf{k})$ (or $H(\mathbf{k})$) to block-diagonal form, one block per irrep — turning a $3N\times 3N$ ($M\times M$) diagonalization into a set of small ones and labelling every eigenvector by its transformation properties for free. At $\Gamma$ the three acoustic modes are the polar-vector irrep ${\Gamma }_{\text{vec}}$ (the translational representation), which the acoustic sum rule pins to $\omega =0$; the optical modes fill out the remainder of ${\Gamma }_{\Gamma }^{\text{mech}}$, and their infrared/Raman activity is just whether $\mu$ matches a vector/quadratic-form irrep.

Time reversal and antiunitary corepresentations

One caveat completes the picture. Time reversal $\Theta$ is antiunitary and acts as $\mathbf{k}\to -\mathbf{k}$, so it relates ${\mathcal{G}}_{\mathbf{k}}$ to ${\mathcal{G}}_{-\mathbf{k}}$ and can glue otherwise distinct small irreps into a single physical corepresentation. Which of three cases obtains is fixed by the Herring (Wigner–Dimmock) criterion — the antiunitary, corepresentation analogue of the Frobenius–Schur reality test. It sums the unitary small-irrep character ${\chi }_{\mathbf{k}}$ not over the little group but over its antiunitary elements $a=\{\Theta R|\dots \}$ — those built from $\Theta$ composed with the spatial operations $R$ that send $\mathbf{k}\to -\mathbf{k}$ (mod $\mathbf{G}$), for which $a^2$ lands back in the unitary little group ${\mathcal{P}}_{\mathbf{k}}$:

$$\frac{1}{|{\mathcal{P}}_{\mathbf{k}}|}\sum _{a\text{antiunitary}}{\chi }_{\mathbf{k}}\left(a^2\right)=\{\begin{array}{ll}+1& \text{(a) real: }{D}_{\mathbf{k}}\cong D_{\mathbf{k}}^{\ast }\text{ — no extra degeneracy;}\\ 0& \text{(b) complex: }{D}_{\mathbf{k}}\text{ glued to a distinct }{D}_{\mathbf{k}}^{\ast };\\ -1& \text{(c) pseudoreal (quaternionic): }{D}_{\mathbf{k}}\text{ doubled,}\end{array}$$

with $|{\mathcal{P}}_{\mathbf{k}}|$ the order of the unitary little co-group (the antiunitary coset has the same number of elements). Writing the sum instead as ${\sum }_g\chi (g^2)$ over the unitary ${\mathcal{P}}_{\mathbf{k}}$ would be the ordinary Frobenius–Schur indicator — the self-reality of $D_{\mathbf{k}}$ alone, blind to the $\Theta$-induced gluing. For phonons (spinless, ${\Theta }^2=+1$) it forces the extra degeneracy $\omega (\mathbf{k})=\omega (-\mathbf{k})$ and can merge complex-conjugate irreps at $\Gamma$; for electrons with spin–orbit coupling (${\Theta }^2=-1$) it is the source of Kramers degeneracy and of the doubled little-group (double-group) representations one must use. I flag it here because it changes the counted degeneracies and the allowed contacts — and so it is part of the same compatibility/star analysis, not an afterthought.

Prerequisites

elementary group theory (groups, subgroups, cosets, conjugacy classes; representations, characters, the orthogonality and reduction formulae, Schur’s lemma) · Bloch’s theorem and the reciprocal lattice / Brillouin zone (the irreps of the translation group)

Builds toward: dynamical matrix · finite-displacement phonons · tight-binding & Bloch’s theorem · chiral phonons

Key references

• The little-group / group-of-the-wavevector method — L. P. Bouckaert, R. Smoluchowski & E. Wigner, Phys. Rev. 50, 58 (1936).
• Comprehensive treatments — C. J. Bradley & A. P. Cracknell, The Mathematical Theory of Symmetry in Solids (Oxford University Press, 1972) (space groups, stars, little groups, corepresentations); T. Inui, Y. Tanabe & Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1990); G. F. Koster, J. O. Dimmock, R. G. Wheeler & H. Statz, Properties of the Thirty-Two Point Groups (MIT Press, 1963) (character and coupling tables).
• Computational tools — M. I. Aroyo et al., the Bilbao Crystallographic Server (Wyckoff positions, $k$-vector tables, compatibility relations, band/representation analysis); Z. Kristallogr. 221, 15 (2006) and Acta Cryst. A 62, 115 (2006).
• Time reversal — C. Herring, Phys. Rev. 52, 361 (1937).