Empty Lattice Model for the Simple Cubic:
A Group Theory Perspective

§1 Free Electrons in a Periodic Box

The simplest conceivable model of an electron in a crystal is one where the ionic potential is ignored entirely. Despite its apparent triviality, the empty lattice model is the essential zeroth-order starting point for nearly-free-electron (NFE) band theory: once we understand the symmetry-enforced structure of the empty-lattice bands, we can treat the periodic potential as a perturbation that opens gaps at the zone boundaries.

The energy eigenvalues for a Bloch wavevector $\mathbf{k}$ in the first Brillouin zone, labelled by a reciprocal lattice vector $\mathbf{K}$, are simply the kinetic energies of plane waves shifted by $\mathbf{K}$:

Each choice of $\mathbf{K}$ defines one "branch" of the band structure; different choices of $\mathbf{K}$ can yield the same energy at a given $\mathbf{k}$, creating the degeneracies we will systematically decompose. The band index $n$ in standard notation corresponds to ordering these branches by energy.

§2 The Simple Cubic Lattice and Its Reciprocal

We specialise to the simple cubic (SC) crystal structure with lattice constant $a$. Its primitive direct lattice vectors are:

The reciprocal lattice is itself simple cubic, with primitive vectors:

satisfying $\mathbf{a}_i \cdot \mathbf{b}_j = 2\pi\delta_{ij}$. A general direct-lattice point and a general reciprocal lattice vector are, respectively:

where $l_i, n_i \in \mathbb{Z}$. A general Bloch wavevector within the first Brillouin zone is written as:

where the reduced coordinates $\mu_i \in [-\tfrac{1}{2}, \tfrac{1}{2})$ are not required to be integers — they parameterise the continuous interior of the Brillouin zone. Substituting into Eq. \eqref{eq:el_1} and using $|\mathbf{k}+\mathbf{K}|^2 = (2\pi/a)^2[(n_1+\mu_x)^2+(n_2+\mu_y)^2+(n_3+\mu_z)^2]$, we arrive at:

For the purpose of classifying degeneracies by symmetry, it is convenient to work with the dimensionless reduced energy

which we use throughout the symmetry analysis below. The overall energy scale $\hbar^2(2\pi)^2/2ma^2$ sets the width of the lowest band.

§3 Zone Folding and Degeneracy Structure

Zone folding is the process of taking the infinite free-electron parabola and mapping every state back into the first Brillouin zone by subtracting an appropriate reciprocal lattice vector $\mathbf{K}$. Wherever two or more branches — labelled by different $\mathbf{K}$ vectors — have the same energy at the same $\mathbf{k}$, we have a band crossing or degeneracy.

At a general point $\mathbf{k}$ in the zone, these crossings are accidental and will typically be lifted by any real potential. However, at high-symmetry points, crossings are enforced by the little group of $\mathbf{k}$ — the subgroup of the full point group $G$ that leaves $\mathbf{k}$ invariant. States belonging to different irreducible representations (irreps) of this little group cannot hybridise, and their crossing is a symmetry-protected degeneracy.

§4 Symmetry Analysis: The Character-Projection Method

To decompose an $N$-fold degenerate level into irreps, we treat the $N$ degenerate plane-wave states $\{|\mathbf{k}+\mathbf{K}_i\rangle\}$ as a basis for a reducible representation $\Gamma_{\rm red}$ of the little group $G_{\mathbf{k}}$. The character of each symmetry operation $g \in G_{\mathbf{k}}$ in this representation is simply:

States that are merely permuted by $g$ contribute 0 to the character (their diagonal matrix element vanishes). The multiplicity $n_\alpha$ of each irrep $\Gamma^{(\alpha)}$ in the decomposition is then given by the Great Orthogonality Theorem:

where $\chi^{(\alpha)}(g)$ is the character of $g$ in the irrep $\Gamma^{(\alpha)}$, read directly from the character table, and $|G_{\mathbf{k}}|$ is the order of the little group.

§5 The $\Gamma$-Point $(0,0,0)$ — Little Group $O_h$

At the zone centre $\boldsymbol{\mu} = (0,0,0)$, the little group is the full cubic point group $O_h$ ($m\bar{3}m$) of order 48. The reduced energy simplifies to:

The lowest shells of reciprocal lattice points, ordered by $\tilde{E}_\Gamma$, are:

• $\tilde{E}_\Gamma = 0$: 1-fold — the single point $(0,0,0)$.
• $\tilde{E}_\Gamma = 1$: 6-fold — all permutations of $(\pm 1, 0, 0)$.
• $\tilde{E}_\Gamma = 2$: 12-fold — all permutations of $(\pm 1, \pm 1, 0)$.
• $\tilde{E}_\Gamma = 3$: 8-fold — all permutations of $(\pm 1, \pm 1, \pm 1)$.

5.1 Significance of the $A_{1g}$ appearance at every level

Notice that $A_{1g}$ (the totally symmetric, one-dimensional irrep) appears in every shell. This is because the $\mathbf{K}=\mathbf{0}$ state — which always exists at the zone centre from the $\Gamma$ branch itself — always transforms as $A_{1g}$. The other irreps in each shell arise from the symmetrised combinations of the remaining degenerate states.

§6 The X-Point $(\tfrac{1}{2},0,0)$ — Little Group $D_{4h}$

Moving to the face-centre of the Brillouin zone along $\hat{k}_x$, the little group reduces from $O_h$ to $D_{4h}$ (order 16), because the $C_4$ axis along $\hat{k}_x$ is preserved but the three-fold $C_3$ axes are broken. The reduced energy becomes:

The lowest degenerate shells are:

• $\tilde{E}_X = \tfrac{1}{4}$: 2-fold — states $(0,0,0)$ and $(-1,0,0)$. These are the two zone-boundary states connected by a $\mathbf{K}=\mathbf{b}_1$ translation.
• $\tilde{E}_X = \tfrac{5}{4}$: 8-fold — states such as $(0,\pm 1,0)$, $(0,0,\pm 1)$, $(-1,\pm 1,0)$, $(-1,0,\pm 1)$.
• $\tilde{E}_X = \tfrac{9}{4}$: 10-fold — including $(1,0,0)$, $(-2,0,0)$, and $(0,\pm 1,\pm 1)$ families.
• $\tilde{E}_X = \tfrac{13}{4}$: 8-fold — including $(\pm 1,\pm 1,0)$ and $(\pm 1,0,\pm 1)$ families shifted by $-\tfrac{1}{2}$.

§7 The R-Point $(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})$ — Little Group $O_h$

The corner of the simple cubic Brillouin zone at $\boldsymbol{\mu} = (\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})$ regains the full $O_h$ symmetry. Physically, this is because the R-point is equally distant from all six face centres and all eight neighbouring corners — the full octahedral symmetry is restored. The reduced energy is:

7.1 Lowest level: $\tilde{E}_R = \tfrac{3}{4}$, 8-fold

Applying the character projection: under $E$ all 8 are fixed ($\chi=8$); under $C_3$ (body diagonal rotation) cycles of length 3 appear among the 6 non-diagonal states, leaving only 2 fixed; under $C_2$ (face diagonal) none are fixed... Evaluating the full set and projecting onto $O_h$ irreps gives:

Dimensions check: $1+1+3+3 = 8$ ✓.

7.2 Second shell: $\tilde{E}_R = \tfrac{11}{4}$, 24-fold

The next shell at $\tilde{E}_R = \tfrac{11}{4}$ arises from all permutations of $(n_1,n_2,n_3)$ containing one entry equal to $+1$ and two entries in $\{0,-1\}$, plus permutations with one entry equal to $-2$. These 24 states form a larger orbit of $O_h$. The irrep decomposition is:

Dimensions check: $1+1+2+2+3+6+6+3 = 24$ ✓.

7.3 Third shell: $\tilde{E}_R = \tfrac{19}{4}$, 24-fold

Remarkably, the third shell at $\tilde{E}_R = \tfrac{19}{4}$ is also 24-fold degenerate and shares the identical irrep decomposition as the second shell:

This is not a coincidence — both 24-fold shells arise from the same orbit structure under $O_h$ (a set of 24 points related by the 24 proper rotations of the cube), and consequently they carry identical symmetry content. The energy difference $\tfrac{19}{4} - \tfrac{11}{4} = 2$ simply reflects the different radii of the two shells in reciprocal space.

7.4 Summary: High-Symmetry Point Analysis

§8 Physical Consequences and Beyond the Empty Lattice

The empty-lattice analysis establishes the symmetry skeleton of the full band structure. When a periodic ionic potential $V(\mathbf{r})$ is switched on:

• Crossings between states of different irreps become avoided crossings (gaps open). The gap magnitude is set by the relevant Fourier component of $V$.
• Crossings between states of the same irrep are generally avoided too — they hybridise and repel. Only crossings at high-symmetry points, protected by the little group, can survive as true degeneracies (Dirac-like touchings).
• The number of bands crossing the Fermi level determines whether the material is a metal or insulator, following the same counting as in the tight-binding and Kronig–Penney pictures.

8.1 Connection to Nearly-Free-Electron Theory

In the nearly-free-electron (NFE) limit, the energy correction to the empty-lattice bands from a weak periodic potential $V$ is calculated by degenerate perturbation theory within each degenerate subspace. The $2\times 2$ secular determinant for states connected by a single reciprocal lattice vector $\mathbf{K}$ gives:

where $\bar{E} = \tfrac{1}{2}(E_{\mathbf{K}}+E_{\mathbf{K}'})$ is the average energy of the two degenerate states and $\delta = \tfrac{1}{2}(E_{\mathbf{K}}-E_{\mathbf{K}'})$ is their energy difference (zero exactly at the zone boundary). The gap $\Delta E = 2|V_\mathbf{K}|$ opens precisely at the crossing point predicted by the empty-lattice analysis, affecting only states with the same $\mathbf{k}$ but different $\mathbf{K}$ labels.

This is why the empty-lattice band structure, combined with the symmetry decomposition into irreps, gives a complete map of where gaps open and how bands reconnect when a potential is added. The group-theory analysis tells us which states mix; the NFE perturbation theory tells us by how much.