Crystalline Symmetry Preview

Crystalline symmetry is what remains of spatial symmetry when a system has a periodic structure. Continuous translations are usually broken, but translations by lattice vectors survive. Continuous rotations are usually reduced to a finite set of rotations, reflections, and inversions compatible with the lattice.

The translation eigenvalue gives crystal momentum, while the symmetry of a state within each momentum sector constrains its degeneracies and allowed couplings. For the geometric construction of cells, lattice bases, and reciprocal vectors, see Crystals and Lattices and Reciprocal Lattice.

From Continuous to Discrete Translations

For a free particle or a constant potential, every spatial translation can be a symmetry. In a crystal, the potential is periodic rather than constant:

$$V(\mathbf r+\mathbf R) = V(\mathbf r), \qquad \mathbf R\in\Lambda,$$

where $\Lambda$ is a lattice of allowed translation vectors.

The continuous translation group is replaced by the discrete translation group $\Lambda$. If $T(\mathbf R)$ translates a state by a lattice vector, the symmetry condition is

$$T(\mathbf R)H T(\mathbf R)^\dagger = H \qquad \text{for every } \mathbf R\in\Lambda.$$

Equivalently,

$$[H,T(\mathbf R)]=0 \qquad \text{for every } \mathbf R\in\Lambda.$$

This is weaker than continuous translation invariance. In a periodic potential, the ordinary momentum operator $\mathbf P$ need not commute with $H$, but the discrete translation operators can.

Lattice Vectors

A Bravais lattice in $d$ dimensions is generated by primitive vectors $\mathbf a_1,\ldots,\mathbf a_d$:

$$\Lambda = \left\{ \mathbf R = \sum_{j=1}^d n_j\mathbf a_j : n_j\in\mathbb Z \right\}.$$

In a real crystal there may also be a basis: several atoms or orbitals attached to each lattice point. The lattice describes the repeating translation pattern; the basis describes what is repeated.

That separation matters. A honeycomb lattice, for example, is often described as a triangular Bravais lattice with a two-site basis. Treating the visible sites as the Bravais lattice can give the wrong translation group and the wrong reciprocal-space labels.

Translation Eigenstates and Bloch Phases

The lattice translations commute with one another:

$$T(\mathbf R)T(\mathbf R') = T(\mathbf R+\mathbf R') = T(\mathbf R')T(\mathbf R).$$

Because the discrete translation group is abelian, states can be chosen to diagonalize all lattice translations at once, within the usual spectral and boundary-condition assumptions. A translation eigenstate has phases

$$T(\mathbf R)|\psi_{\mathbf k}\rangle = e^{-i\mathbf k\cdot\mathbf R} |\psi_{\mathbf k}\rangle.$$

With the active convention for wavefunctions,

$$(T(\mathbf R)\psi)(\mathbf r) = \psi(\mathbf r-\mathbf R).$$

Therefore the translation eigenvalue condition becomes

$$\psi_{\mathbf k}(\mathbf r+\mathbf R) = e^{i\mathbf k\cdot\mathbf R} \psi_{\mathbf k}(\mathbf r).$$

This is the phase law behind Bloch waves. It implies the familiar form

$$\psi_{n\mathbf k}(\mathbf r) = e^{i\mathbf k\cdot\mathbf r} u_{n\mathbf k}(\mathbf r),$$

where

$$u_{n\mathbf k}(\mathbf r+\mathbf R) = u_{n\mathbf k}(\mathbf r).$$

The label $n$ distinguishes bands or other independent states with the same crystal momentum.

Reciprocal-Lattice Equivalence

The reciprocal lattice $\Lambda^*$ consists of vectors $\mathbf G$ satisfying

$$e^{i\mathbf G\cdot\mathbf R}=1 \qquad \text{for every } \mathbf R\in\Lambda.$$

Thus $\mathbf k$ and $\mathbf k+\mathbf G$ give the same translation eigenvalues:

$$e^{i(\mathbf k+\mathbf G)\cdot\mathbf R} = e^{i\mathbf k\cdot\mathbf R}.$$

Crystal momentum is therefore not an ordinary vector momentum with a unique value in all of reciprocal space. It is defined modulo reciprocal-lattice vectors:

$$\mathbf k\sim\mathbf k+\mathbf G.$$

Brillouin Zones develops the Wigner–Seitz construction, boundary identifications, zone schemes, and high-symmetry paths used to choose representatives. The reciprocal-lattice algebra itself is developed in Poisson Summation Formula and the Fourier-series background in Periodic Functions and Fourier Series.

One-Dimensional Periodic Potential

For a one-dimensional potential with period $a$,

$$V(x+a)=V(x),$$

the symmetry is generated by a single discrete translation $T(a)$. Translation eigenstates obey

$$T(a)|\psi_k\rangle=e^{-ika}|\psi_k\rangle,$$

or in position space

$$\psi_k(x+a)=e^{ika}\psi_k(x).$$

The reciprocal-lattice wave numbers are

$$G_m=\frac{2\pi m}{a}, \qquad m\in\mathbb Z,$$

so $k$ and $k+2\pi/a$ label the same translation character.

For a finite ring with $N$ unit cells and total length $L=Na$, periodic boundary conditions further quantize $k$:

$$k_m=\frac{2\pi m}{Na}.$$

In the large-system limit these allowed values become dense inside a Brillouin zone.

Point Groups and Space Groups

Lattice translations are only part of crystalline symmetry. The crystallographic point group consists of the orthogonal parts of space-group operations, after translations are factored out. For a symmorphic crystal these can all be represented by operations fixing a common point. In a nonsymmorphic crystal, a point-group element may occur only together with a fractional translation.

For example:

• a square lattice is compatible with rotations by multiples of $\pi/2$;
• a rectangular lattice is generally compatible with fewer rotations;
• inversion may be present or absent depending on the lattice plus basis;
• a basis can remove a symmetry that the underlying Bravais lattice would otherwise have.

The point group acts on crystal momentum labels by sending $\mathbf k$ to $g\mathbf k$, modulo reciprocal-lattice vectors. At generic $\mathbf k$, this may move the state to a different momentum sector. At special momenta, the little group of $\mathbf k$ can constrain degeneracies, matrix elements, and band crossings.

A full space group combines translations with point-group operations. Some crystals also have nonsymmorphic symmetries, such as glides and screw rotations, which combine a point operation with a fractional translation. Symmetry of Bloch States explains how these operations act on Bloch states and constrain degeneracies. The space group need not be a simple direct product of translations and the point group.

Momentum Conservation Modulo a Reciprocal Vector

Because only discrete translations are exact, the conserved label is crystal momentum modulo reciprocal-lattice vectors. A translation-invariant interaction in a perfect lattice can conserve total crystal momentum in the form

$$\mathbf k_f = \mathbf k_i+\mathbf q+\mathbf G,$$

where $\mathbf q$ is the wave vector carried by the perturbation and $\mathbf G$ is a reciprocal-lattice vector.

The reciprocal vector expresses equality of translation eigenvalues. In scattering among crystal excitations, a nonzero reciprocal-vector transfer, with momenta represented in a chosen first Brillouin zone, is called an Umklapp process. Merely relabeling one Bloch state by $\mathbf k+\mathbf G$ is not a scattering event.

This statement should not be confused with conservation of mechanical momentum. In a periodic potential, the lattice background has already broken continuous translation symmetry.

Assumptions and Limits

The clean crystalline symmetry picture assumes:

• an ideal infinite or periodically repeated lattice;
• a Hamiltonian invariant under the stated lattice translations and point operations;
• boundary conditions compatible with those symmetries;
• no disorder, defects, surfaces, strain, or time-dependent driving that destroys the symmetry being used;
• a single-particle or effective quasiparticle setting when using simple Bloch-state language.

Finite samples, open boundaries, disorder, interactions, magnetic fields, and strong correlations can keep part of the language while changing the exact symmetry content. The right question is not “is there a lattice?” but “which transformations commute with the Hamiltonian and its domain?”

Exercises

1. In one dimension, suppose $T(a)|\psi_k\rangle=e^{-ika}|\psi_k\rangle$ and $(T(a)\psi)(x)=\psi(x-a)$. Show that $\psi_k(x+a)=e^{ika}\psi_k(x)$.

Solution

The eigenvalue equation in position space is

$$\psi_k(x-a) = e^{-ika}\psi_k(x).$$

Replace $x$ by $x+a$:

$$\psi_k(x) = e^{-ika}\psi_k(x+a).$$

Multiplying by $e^{ika}$ gives

$$\psi_k(x+a) = e^{ika}\psi_k(x).$$

2. For a one-dimensional lattice of spacing $a$, show that $k$ and $k+2\pi/a$ give the same lattice-translation phase.

Solution

For a lattice vector $R=na$,

$$e^{i(k+2\pi/a)R} = e^{ikna}e^{i2\pi n} = e^{ikna}.$$

Since $e^{i2\pi n}=1$ for every integer $n$, the two wave numbers define the same character of the translation group.

3. On a periodic chain with $N$ sites and spacing $a$, take $k=2\pi m/(Na)$ and define

$$|k\rangle = \frac{1}{\sqrt N} \sum_{j=0}^{N-1} e^{ikja}|j\rangle.$$

If $T(a)|j\rangle=|j+1\rangle$ with site labels understood modulo $N$, show that $T(a)|k\rangle=e^{-ika}|k\rangle$.

Solution

Apply the translation:

$$T(a)|k\rangle = \frac{1}{\sqrt N} \sum_{j=0}^{N-1} e^{ikja}|j+1\rangle.$$

Let $\ell=j+1$ modulo $N$. The wrapped endpoint has the same coefficient because $e^{ikNa}=1$; without this quantization the proposed vector is not a translation eigenstate. Thus

$$T(a)|k\rangle = \frac{1}{\sqrt N} \sum_{\ell=0}^{N-1} e^{ik(\ell-1)a}|\ell\rangle = e^{-ika}|k\rangle.$$

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976.
• International Union of Crystallography, “Definition of symmetry elements in space groups and point groups,” nomenclature report, 1989. Report.
• C. Kittel, Introduction to Solid State Physics, 8th ed., Wiley, 2004.
• M. P. Marder, Condensed Matter Physics, 2nd ed., Wiley, 2010.
• S. H. Simon, The Oxford Solid State Basics, Oxford University Press, 2013.
• M. Tinkham, Group Theory and Quantum Mechanics, Dover, 2003 reprint of the 1964 edition.