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.

This page is a preview. It explains the symmetry skeleton behind lattice translations, Bloch phases, reciprocal-lattice equivalence, and point-group constraints. Crystals and Lattices is the canonical home for the direct-space definitions, cell conventions, worked geometries, and ideal-lattice approximation. Long-Range Order explains how spontaneous crystalline order appears through density correlations and Bragg scaling, while Goldstone Modes in Many-Body Systems explains why broken translations yield acoustic phonons without an independent mode for every broken rotation. The Tight-Binding Model shows how an imposed periodic skeleton diagonalizes a concrete hopping Hamiltonian. Mathematical preparation is collected in Math Needed for Quantum Matter.

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

The canonical distinction between a translation lattice and its repeated motif is developed in Crystals and Lattices. For this symmetry preview, 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

Reciprocal Lattice is the canonical home for the phase-invariance definition, reciprocal-basis construction, diffraction condition, and momentum selection rules. In compact symmetry language, 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. A point-group operation is a rotation, reflection, inversion, or rotoinversion that leaves at least one point fixed and maps the lattice to itself.

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 owns their Bloch-fiber representations, little-group labels, compatibility relations, and degeneracy or crossing constraints. This preview stops at the general group skeleton: always state the group action before assigning labels.

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 appearance of $\mathbf G$ is not a bookkeeping trick. It says the lattice can absorb reciprocal-lattice momentum while preserving the discrete translation symmetry. In solid-state language, such processes are often called Umklapp processes.

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?”

Common Mistakes

• Treating crystal momentum as ordinary mechanical momentum.
• Applying Bloch phases when the Hamiltonian or boundary conditions are not lattice-periodic.
• Confusing the Bravais lattice with the basis.
• Forgetting that $\mathbf k$ and $\mathbf k+\mathbf G$ label the same translation character.
• Assuming a point-group symmetry of the Bravais lattice survives after adding a basis or external field.
• Using continuous-rotation or continuous-translation labels in a crystal without checking which continuous symmetries are actually broken.

Cross-Links

• Translations and Momentum
• Translation-Invariant Hamiltonians
• Rotations Preview
• Parity as Spatial Inversion
• Group Actions
• Unitary Representations
• Periodic Functions and Fourier Series
• Poisson Summation Formula
• Periodic Boundary Conditions
• Math Needed for Quantum Matter

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976.
• C. Kittel, Introduction to Solid State Physics, 8th ed., Wiley, 2004.
• S. H. Simon, The Oxford Solid State Basics, Oxford University Press, 2013.
• M. P. Marder, Condensed Matter Physics, 2nd ed., Wiley, 2010.
• M. Tinkham, Group Theory and Quantum Mechanics, Dover, 2003.

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 $a$, 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 $N$. Then $j=\ell-1$, so

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