Brillouin Zones

The first Brillouin zone is the Wigner–Seitz cell of the reciprocal lattice around $\mathbf k=\mathbf0$. It is a symmetry-adapted choice of one representative from each equivalence class

$$\mathbf k \sim \mathbf k+\mathbf G, \qquad \mathbf G\in\Lambda^*.$$

The zone is therefore both a geometric polytope and a fundamental domain for crystal momentum. Its faces are perpendicular bisectors to reciprocal-lattice vectors, its opposite boundary pieces are identified, and its volume is fixed by the direct primitive cell.

Bloch’s Theorem constructs the states labeled by these crystal momenta. Here we develop the zone geometry, explain the different ways to plot bands, and show why a one-dimensional path cannot establish every property of a higher-dimensional band.

Required background. Reciprocal Lattice supplies reciprocal vectors, reciprocal metrics, and the equivalence $\mathbf k\sim\mathbf k+\mathbf G$ used to construct a zone.

First Brillouin Zone

Let $\Lambda^*$ be the reciprocal lattice. The closed first Brillouin zone is

$$\mathcal B_1 = \left\{ \mathbf k\in\mathbb R^d \;\middle|\; |\mathbf k| \leq |\mathbf k-\mathbf G| \text{ for every } \mathbf G\in\Lambda^* \right\}.$$

It contains the points at least as close to the reciprocal origin as to any other reciprocal-lattice point. Expanding the squared distances gives

$$|\mathbf k|^2 \leq |\mathbf k-\mathbf G|^2 = |\mathbf k|^2 - 2\mathbf k\cdot\mathbf G + |\mathbf G|^2.$$

Thus

$$\mathcal B_1 = \bigcap_{\mathbf G\neq\mathbf0} \left\{ \mathbf k \;\middle|\; \mathbf k\cdot\mathbf G \leq \frac{|\mathbf G|^2}{2} \right\}.$$

Each nonzero $\mathbf G$ defines a half-space bounded by the perpendicular-bisector plane

$$\mathbf k\cdot\mathbf G = \frac{|\mathbf G|^2}{2}.$$

Only a finite set of nearby reciprocal vectors contributes faces. More distant bisectors lie outside the polytope already enclosed by nearer ones.

Construction algorithm

To construct the first zone geometrically:

1. draw the reciprocal-lattice origin and a sufficient shell of neighboring reciprocal points;
2. join the origin to each neighbor;
3. draw the perpendicular bisector of each joining segment;
4. keep the intersection of the half-spaces containing the origin;
5. verify that the enclosed measure equals one primitive reciprocal-cell measure.

The result is basis independent. A reciprocal primitive parallelepiped is also a valid fundamental domain, but its faces need not expose the full reciprocal point symmetry. The Wigner–Seitz choice is preferred when discussing degeneracies, symmetry lines, and diffraction planes.

Volume

Every primitive reciprocal cell has measure

$$\Omega_{\mathrm{BZ}} = \frac{(2\pi)^d}{\Omega_{\mathrm c}},$$

where $\Omega_{\mathrm c}$ is the direct primitive-cell measure. Therefore

$$\operatorname{vol}(\mathcal B_1) = \Omega_{\mathrm{BZ}}.$$

This identity is a stringent numerical check. A purported first zone with another volume is missing a region, double-counting one, or built from the wrong direct translation cell.

One Representative Per Translation Character

Translations act on a crystal-momentum state through phases

$$T_{\mathbf R} \longmapsto e^{-i\mathbf k\cdot\mathbf R}$$

with the active operator convention. Since

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

$\mathbf k$ and $\mathbf k+\mathbf G$ define the same character of the direct translation group. Reciprocal space is therefore quotiented by $\Lambda^*$:

$$\mathbb T_{\mathrm{BZ}}^d = \mathbb R^d/\Lambda^*.$$

Topologically this quotient is a $d$-torus, even when its Wigner–Seitz drawing is a hexagon, truncated octahedron, or another polytope. Opposite or symmetry-related faces are glued by reciprocal translations. A zone face is not a physical edge of a crystal-momentum world.

Interior and boundary representatives

Every equivalence class has one representative in a half-open first zone. Boundary classes have two or more representatives in the closed Wigner–Seitz polytope. For example, in one dimension,

$$-\frac{\pi}{a} \sim \frac{\pi}{a}$$

because the two endpoints differ by $2\pi/a$.

Boundary duplication has measure zero in integrals but matters in finite sums, mesh generation, interpolation, and file formats. A numerical implementation must choose a half-open convention or merge equivalent points explicitly.

Zone Boundaries as Bragg Planes

A face associated with $\mathbf G$ satisfies

$$\mathbf k\cdot\mathbf G = \frac{|\mathbf G|^2}{2}.$$

On this plane,

$$|\mathbf k| = |\mathbf k-\mathbf G|.$$

For the free-particle dispersion

$$\varepsilon_0(\mathbf q) = \frac{\hbar^2|\mathbf q|^2}{2m},$$

the plane waves $\mathbf k$ and $\mathbf k-\mathbf G$ are degenerate:

$$\varepsilon_0(\mathbf k) = \varepsilon_0(\mathbf k-\mathbf G).$$

A periodic potential contains Fourier components $V_{\mathbf G}$ that can couple precisely these wavevectors. Near an isolated two-state crossing, and for a potential weak enough that other plane waves give only small corrections, the effective matrix is

$$H_{\mathrm{edge}} = \begin{pmatrix} \varepsilon_0&V_{\mathbf G}\\ V_{\mathbf G}^*&\varepsilon_0 \end{pmatrix},$$

with eigenvalues

$$E_\pm = \varepsilon_0 \pm |V_{\mathbf G}|.$$

Here the constant Fourier component $V_{\mathbf0}$ has been absorbed into the energy origin. The local splitting is $2|V_{\mathbf G}|$. It need not be a global band gap: other states elsewhere in the zone can lie at the same energy. The equal-length scattering condition explains why the zone faces are also Bragg planes.

A gap is not automatic at every apparent crossing. The coupling can vanish because of symmetry, internal structure, polarization, or a zero Fourier component. Degenerate representations can also protect crossings. The systematic weak-potential calculation is developed in Nearly Free Electrons.

One-Dimensional Zones

For a chain with direct period $a$,

$$G_m = \frac{2\pi m}{a}.$$

The first Brillouin zone can be chosen as

$$-\frac{\pi}{a} \leq k < \frac{\pi}{a}.$$

The first boundaries lie halfway between $G_0=0$ and $G_{\pm1}=\pm2\pi/a$. Successive Bragg points partition the line into zones:

$$\begin{aligned} \mathcal B_1 &: \left|k\right| < \frac{\pi}{a}, \\ \mathcal B_2 &: \frac{\pi}{a} < \left|k\right| < \frac{2\pi}{a}, \end{aligned}$$

with boundary assignments supplied separately. Higher-zone terminology is useful in the extended-zone scheme, but all translation characters already have representatives in $\mathcal B_1$.

For $N$ periodically identified cells, the inequivalent mesh is

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

with any $N$ consecutive integer values of $m$. Exactly $N$ points occur in one Brillouin zone.

Square and Triangular Examples

The first zone is bounded by perpendicular bisectors to the nearest reciprocal points. The displayed square path is $\Gamma\!-\!X\!-\!M\!-\!\Gamma$; the triangular path is $\Gamma\!-\!M\!-\!K\!-\!\Gamma$. These labels are explicitly tied to the depicted reciprocal bases and should not be transferred to another convention without checking coordinates.

Square lattice

For a direct square lattice of period $a$,

$$\mathbf b_1 = \frac{2\pi}{a}(1,0), \qquad \mathbf b_2 = \frac{2\pi}{a}(0,1).$$

The first zone is

$$-\frac{\pi}{a} \leq k_x,k_y \leq \frac{\pi}{a},$$

with a half-open boundary convention understood when unique counting is required.

One common coordinate declaration is

$$\begin{aligned} \Gamma&=(0,0),\\ X&=\frac12\mathbf b_1,\\ M&=\frac12(\mathbf b_1+\mathbf b_2). \end{aligned}$$

$\Gamma$ is the zone center, $X$ is an edge midpoint, and $M$ is a corner.

Triangular reciprocal lattice

Choose reciprocal vectors of equal length meeting at $60^\circ$. The first zone is a regular hexagon. For the orientation in the figure, representative special points are

$$\begin{aligned} \Gamma&=\mathbf0,\\ M&=\frac12\mathbf b_1,\\ K&=\frac13(\mathbf b_1+\mathbf b_2). \end{aligned}$$

$M$ is an edge midpoint and $K$ is a corner. Their point-group stars contain all symmetry-equivalent edge midpoints and corners. A honeycomb crystal uses this same hexagonal first zone because its Bravais lattice is triangular; the two-site motif changes the bands, not the reciprocal translation lattice.

Three-Dimensional Shapes

The Wigner–Seitz construction extends directly to three dimensions:

• a simple-cubic direct lattice has a cubic reciprocal lattice and cubic first zone;
• an fcc direct lattice has a bcc reciprocal lattice and a truncated-octahedral first zone;
• a bcc direct lattice has an fcc reciprocal lattice and a rhombic-dodecahedral first zone.

For simple cubic,

$$-\frac{\pi}{a} \leq k_x,k_y,k_z \leq \frac{\pi}{a}.$$

A commonly used path includes

$$\begin{aligned} \Gamma&=(0,0,0),\\ X&=\frac{\pi}{a}(1,0,0),\\ M&=\frac{\pi}{a}(1,1,0),\\ R&=\frac{\pi}{a}(1,1,1). \end{aligned}$$

Noncubic zones can change shape as axial ratios vary even within the same broad lattice family. Automated band-path conventions therefore standardize the conventional cell, primitive basis, and metric case before assigning labels.

Point-Group Symmetry in the Zone

Let $\mathcal P$ be the reciprocal point group. It maps the reciprocal lattice to itself:

$$p\Lambda^* = \Lambda^*, \qquad p\in\mathcal P.$$

It also maps the first zone to itself. The star of $\mathbf k$ is

$$\operatorname{star}(\mathbf k) = \left\{ p\mathbf k \bmod\Lambda^* \;\middle|\; p\in\mathcal P \right\}.$$

All points in a star are symmetry related when the full Hamiltonian has that point-group symmetry.

Little group

The little co-group of $\mathbf k$ consists of operations satisfying

$$p\mathbf k = \mathbf k+\mathbf G_p$$

for some reciprocal vector $\mathbf G_p$. Generic interior points usually have the smallest little group. Zone centers, face centers, edges, corners, and special lines can have larger little groups and therefore stronger constraints on energies and eigenstates.

High-symmetry does not mean “highest energy,” “lowest energy,” or “most important for every observable.” It means that the wavevector is fixed modulo a reciprocal vector by more symmetry operations than a nearby generic point.

Irreducible Brillouin zone

An irreducible Brillouin zone contains one representative from each symmetry star. Use symmetries of the full Hamiltonian and the quantity being integrated; a geometrically square lattice can still have a Hamiltonian that breaks fourfold rotation.

For a continuous, symmetry-invariant scalar integrand and a genuine fundamental wedge $W$ of a point group acting effectively on momentum,

$$\int_{\mathcal B_1} f(\mathbf k)\,d^dk =|\mathcal P|\int_W f(\mathbf k)\,d^dk.$$

Shared wedge boundaries have zero measure, so they do not invalidate this integral relation. Discrete meshes require more care: special points can have shorter stars. For a symmetry-closed mesh of $N_k$ distinct points,

$$\frac{1}{N_k}\sum_{\mathbf k}f(\mathbf k) =\sum_\nu \frac{|\operatorname{star}(\mathbf k_\nu)|}{N_k} f(\mathbf k_\nu).$$

For example, the square $5\times5$ mesh $\mathbf k=(2\pi/5a)(n_x,n_y)$, with $n_x,n_y=-2,\ldots,2$, has six stars under the eight square point-group operations. Their sizes are $1,4,4,4,4,8$, totaling 25. The origin therefore has weight $1/25$, whereas the generic star represented by $(n_x,n_y)=(1,2)$ has weight $8/25$. Assigning every representative weight $8/25$ would count 48 points.

Time reversal can also relate $\mathbf k$ and $-\mathbf k$, when present and when the integrand transforms appropriately. Spinor phases alone do not change scalar integration weights, but magnetic order, spin-orbit coupling and external fields can change the applicable symmetry group. Tensor and matrix integrands must be transformed, not simply copied as scalar values.

High-Symmetry Labels and Paths

$\Gamma$ universally denotes $\mathbf k=\mathbf0$. Other labels such as $X$, $M$, $K$, $L$, $W$, $H$, and $R$ are conventional, not self-defining. Their coordinates can depend on:

• Bravais type and space group;
• conventional-cell setting and primitive basis;
• reciprocal-vector order and orientation;
• axial-ratio regime;
• whether spin, magnetism, or nonsymmorphic symmetry is included;
• the selected path standard.

A reproducible band plot should therefore declare the structure, space group or layer group, direct and reciprocal bases, fractional coordinates of every labeled point, and the path convention.

Authoritative machine-readable standards include the Bilbao Crystallographic Server k-vector database, the AFLOW convention, and the crystallographic convention implemented by SeeK-path. Their labels and recommended paths are not interchangeable in every setting.

What a path coordinate means

For vertices $\mathbf k_0,\mathbf k_1,\ldots$, a plotting coordinate is often accumulated reciprocal-space distance:

$$s_j = \sum_{\ell<j} \left| \mathbf k_{\ell+1} - \mathbf k_\ell \right|.$$

Within a segment,

$$\mathbf k(t) = (1-t)\mathbf k_j + t\mathbf k_{j+1}, \qquad 0\leq t\leq1.$$

The horizontal axis in a band diagram is then path distance $s$, not a Cartesian component such as $k_x$. At a corner where the path direction changes, left and right slopes probe different directional derivatives.

Some plotting tools assign equal visual width to each segment instead of using physical reciprocal distance. The displayed slope then cannot be read directly as a group velocity without reconstructing the actual $\mathbf k$ spacing.

Why Band Plots Use Special Paths

Special paths compress a multidimensional function $E_n(\mathbf k)$ into a readable one-dimensional diagram. They are valuable because:

• little-group representations label states and constrain degeneracies;
• symmetry-enforced crossings often occur on special points or lines;
• common conventions permit qualitative comparison among calculations;
• zone-center and boundary behavior becomes visible;
• computational cost is far lower than a dense full-zone visualization.

But a path is a set of measure zero inside a two- or three-dimensional zone. It can miss:

• a conduction-band minimum or valence-band maximum at generic $\mathbf k$;
• an indirect gap;
• a nodal loop or Weyl point away from the chosen line;
• a small Fermi pocket;
• avoided crossings displaced by symmetry breaking;
• anisotropy away from the plotted directions.

A high-symmetry band plot is not sufficient for a density of states, carrier concentration, Fermi surface, transport tensor, optical integral, Berry-curvature integral, or topological invariant. Those require an appropriate full-zone mesh or an adaptive calculation.

Reduced, Extended, and Repeated Schemes

Three representations of the same free-particle states, with $E_\pi=\hbar^2\pi^2/(2ma^2)$. The extended plot uses the physical wavevector $q$; the reduced plot folds it to $k\in[-\pi/a,\pi/a]$; the repeated plot copies the reduced spectrum. Each panel uses the same energy scale. Black denotes the lowest reduced band and gray the higher branches; only energies up to $4E_\pi$ are shown. Crossings remain ungapped because no periodic potential has been added.

Every wavevector $\mathbf q$ can be written

$$\mathbf q = \mathbf k+\mathbf G, \qquad \mathbf k\in\mathcal B_1.$$

The interior representative $\mathbf k$ is unique. Different plotting schemes decide whether to display $\mathbf q$, $\mathbf k$, or periodic copies of both.

Extended-zone scheme

The extended-zone scheme lets the displayed wavevector pass through successive Brillouin zones. It is useful for seeing how a nearly free branch evolves through Bragg planes and for connecting to scattering kinematics.

For a free particle, it is simply

$$E(\mathbf q) = \frac{\hbar^2|\mathbf q|^2}{2m}.$$

Each state is displayed once, but equivalent translation characters occur in different zones with different reciprocal offsets.

Reduced-zone scheme

The reduced-zone scheme maps every $\mathbf q$ into $\mathcal B_1$. The reciprocal offset becomes a branch or band label:

$$\varepsilon_{\mathbf G}^{(0)}(\mathbf k) = \frac{\hbar^2}{2m} |\mathbf k+\mathbf G|^2.$$

An infinite set of free parabolas is folded into the first zone. A periodic potential couples the folded branches and reorganizes them into bands. The reduced scheme is the natural representation for state counting, Brillouin-zone integration, and multiband Hamiltonians.

Repeated-zone scheme

The repeated-zone scheme copies the reduced-zone spectrum into every reciprocal cell:

$$\{E_n(\mathbf k+\mathbf G)\}_n = \{E_n(\mathbf k)\}_n.$$

The equality is safest as an equality of spectra. Depending on basis embedding, gauge, and degeneracies, individual band labels or Hamiltonian matrices can transform by a reciprocal sewing unitary rather than remain entry-by-entry identical.

Repeated-zone plots are useful for comparing several pockets, visualizing momentum-transfer processes, and overlaying experimental reciprocal-space maps.

These are representations, not approximations

Choosing reduced, extended, or repeated notation does not change the physical eigenstates. Confusing a plotting scheme with an approximation leads to false claims such as “zone folding creates a gap.” Folding changes labels; a perturbation that couples the folded sectors is needed to change eigenvalues.

Reciprocal Sewing and Basis Embedding

In the default cell convention, a Bloch Hamiltonian may be strictly periodic:

$$H(\mathbf k+\mathbf G) = H(\mathbf k).$$

In an orbital-embedded convention, define

$$\bigl[D_{\mathbf G}\bigr]_{\alpha\beta} = \delta_{\alpha\beta} e^{-i\mathbf G\cdot\boldsymbol\tau_\alpha}.$$

Then a common transformation law is

$$H(\mathbf k+\mathbf G) = D_{\mathbf G} H(\mathbf k) D_{\mathbf G}^{\dagger}.$$

The eigenvalues are periodic, while eigenvectors acquire basis-dependent phases. Berry connections and numerical overlaps are sensitive to this sewing convention even though gauge-invariant observables are not.

The signs of these phases follow the cell and orbital Fourier conventions.

State Counting and Zone Integrals

For a finite periodic crystal with $N_{\mathrm c}$ primitive cells, each isolated band contains $N_{\mathrm c}$ crystal-momentum states before spin or other internal multiplicities.

The thermodynamic replacement is

$$\frac{1}{N_{\mathrm c}} \sum_{\mathbf k\in\mathcal B_1} f(\mathbf k) \longrightarrow \frac{\Omega_{\mathrm c}}{(2\pi)^d} \int_{\mathcal B_1} d^dk\, f(\mathbf k).$$

Because

$$\frac{\Omega_{\mathrm c}}{(2\pi)^d} \operatorname{vol}(\mathcal B_1) = 1,$$

a constant integrand averages to itself. In a sample of physical measure $V=N_{\mathrm c}\Omega_{\mathrm c}$,

$$\sum_{\mathbf k} \longrightarrow \frac{V}{(2\pi)^d} \int_{\mathcal B_1} d^dk.$$

These normalizations are the bridge from band sums to particle density, total energy, response, and density of states.

Mesh choices

A uniform mesh can be written

$$\mathbf k_{\mathbf n} = \sum_{i=1}^{d} \frac{n_i+\eta_i}{N_i} \mathbf b_i,$$

where offsets $\eta_i$ specify centered or shifted sampling. A symmetry-reduced mesh assigns each irreducible point a weight equal to the number of represented full-zone points, adjusted at boundaries.

Convergence must be tested for the observable. Metals, small pockets, sharp Berry-curvature features, van Hove singularities, and near-degenerate bands can require much denser or adaptive meshes than a gapped total-energy calculation.

Zone Folding From Supercells and Order

Suppose a new direct cell contains

$$q=|\det S|$$

primitive cells:

$$A_{\mathrm s} = AS.$$

Its reciprocal basis is

$$B_{\mathrm s} = B S^{-\mathsf T},$$

and its Brillouin-zone volume is

$$\Omega_{\mathrm{BZ}}^{(\mathrm s)} = \frac{1}{q} \Omega_{\mathrm{BZ}}.$$

The smaller supercell zone receives $q$ folded branches for each primitive-cell band.

Two situations must be distinguished:

1. Computational supercell only. The physical Hamiltonian still has primitive translations. Folding is a redundant representation, and primitive crystal momentum remains a hidden exact label.
2. Physical enlarged period. Structural, magnetic, charge, or orbital order genuinely reduces translation symmetry. States separated by new reciprocal vectors can hybridize, and new gaps or reconstructed pockets may appear.

Band unfolding estimates how much primitive-cell spectral character a supercell eigenstate carries. It is a projection with weights, not an exact restoration of a symmetry that the physical system has broken.

When the Ordinary Zone Needs Modification

The ordinary Brillouin zone assumes exact discrete translations.

Situation	Appropriate modification
commensurate superstructure	use the enlarged real cell and folded zone
antiferromagnetic or density-wave order	use the translation group of the ordered phase
uniform magnetic field with rational flux	use magnetic translations and a magnetic Brillouin zone
surface or interface	preserve only translations parallel to the surface and use a projected surface zone
finite open sample	crystal momentum is approximate away from boundaries
disorder	use spectral functions, configuration averages, or unfolded momentum weight
incommensurate structure	use approximants or higher-dimensional formulations with declared limits
quasicrystal	no ordinary finite reciprocal primitive cell exists

In an interacting many-body system, total crystal momentum still labels translation sectors when symmetry is exact, but a single-particle band $E_n(\mathbf k)$ need not describe the excitations. Zone geometry survives more generally than independent-electron band theory.

Exercises

Exercise 1: square-zone coordinates

For a direct square lattice of period $a$, verify the coordinates of $\Gamma$, $X$, and $M$ above. Compute the path lengths $|\Gamma X|$, $|XM|$, and $|M\Gamma|$.

Solution

The reciprocal vectors are

$$\mathbf b_1 = \frac{2\pi}{a}(1,0), \qquad \mathbf b_2 = \frac{2\pi}{a}(0,1).$$

Thus

$$X=\frac{\pi}{a}(1,0), \qquad M=\frac{\pi}{a}(1,1).$$

The segment lengths are

$$|\Gamma X| = \frac{\pi}{a}, \qquad |XM| = \frac{\pi}{a}, \qquad |M\Gamma| = \frac{\sqrt2\pi}{a}.$$

A band plot using physical cumulative distance should therefore draw the diagonal segment longer than either edge segment.

Exercise 2: doubling a one-dimensional cell

A one-dimensional Hamiltonian initially has period $a$. It is represented in a cell of length $2a$. Find the new zone and identify which primitive-zone wavevectors fold together.

Solution

The primitive reciprocal vector is

$$G = \frac{2\pi}{a}.$$

The doubled-cell reciprocal vector is

$$G_{\mathrm s} = \frac{2\pi}{2a} = \frac{\pi}{a}.$$

The new first zone is

$$-\frac{\pi}{2a} \leq k < \frac{\pi}{2a}.$$

Primitive-zone labels separated by $G_{\mathrm s}=\pi/a$ map to the same supercell label:

$$k \quad\text{and}\quad k+\frac{\pi}{a}.$$

If $2a$ is only a computational choice, the two folded sectors cannot hybridize under a Hamiltonian that still respects translation by $a$. If a real dimerization breaks the $a$ translation, coupling between them is allowed and a gap can open at the new boundary.

Exercise 3: a band plot that misses a Fermi crossing

Set $a=1$ and define two square-lattice dispersions, with $E_0>0$,

$$\begin{aligned} E_1(\mathbf k)&=E_0,\\ E_2(\mathbf k)&=E_0[1-8f(\mathbf k)],\\ f(\mathbf k)&=\sin^2 k_x\sin^2 k_y (\cos k_x-\cos k_y)^2. \end{aligned}$$

Show that they agree on $\Gamma\!\to X\!\to M\!\to\Gamma$, with $X=(\pi,0)$ and $M=(\pi,\pi)$, although only the second band crosses the chemical potential $\mu=0$. Does the second dispersion preserve the square symmetries?

Solution

On $\Gamma X$, $k_y=0$; on $XM$, $k_x=\pi$; and on $M\Gamma$, $k_x=k_y$. In each case one factor of $f$ vanishes. Both plots therefore show the constant energy $E_0$ along the entire path.

At the off-path point $(\pi/2,\pi/3)$, however,

$$f=1\times\frac34\times\frac14=\frac{3}{16}, \qquad E_2=-\frac{E_0}{2}.$$

Continuity forces a zero of $E_2$ between this point and the positive-energy path. Reflections and interchange of $k_x,k_y$ leave $f$ invariant, so this counterexample has all square point-group symmetries. It is also periodic and a finite trigonometric polynomial, hence realizable as a one-orbital model with finite-range hoppings. Full symmetry does not make a special path sufficient to locate every Fermi crossing.

Exercise 4: sewing across a reciprocal boundary

Show that the orbital-embedded transformation

$$H(\mathbf k+\mathbf G) = D_{\mathbf G} H(\mathbf k) D_{\mathbf G}^{\dagger}$$

preserves eigenvalues. What information can still change?

Solution

$D_{\mathbf G}$ is diagonal and unitary because each entry has unit magnitude:

$$D_{\mathbf G}^{\dagger}D_{\mathbf G} = I.$$

Unitary conjugation preserves the characteristic polynomial:

$$\det \left[ E- D_{\mathbf G}H D_{\mathbf G}^{\dagger} \right] = \det \left[ D_{\mathbf G}(E-H)D_{\mathbf G}^{\dagger} \right] = \det(E-H).$$

Thus the eigenvalue set is unchanged. Eigenvector components, orbital phases, Berry connections, and raw overlap matrices can change. Gauge-covariant or gauge-invariant combinations must include the sewing unitary when crossing the zone boundary.

References

• M. I. Aroyo, D. Orobengoa, G. de la Flor, E. S. Tasci, J. M. Perez-Mato, and H. Wondratschek, “Brillouin-zone database on the Bilbao Crystallographic Server,” Acta Crystallographica A 70, 126–137 (2014), doi:10.1107/S205327331303091X.
• N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976), Chapters 5–9.
• Y. Hinuma, G. Pizzi, Y. Kumagai, F. Oba, and I. Tanaka, “Band structure diagram paths based on crystallography,” Computational Materials Science 128, 140–184 (2017), doi:10.1016/j.commatsci.2016.10.015.
• H. J. Monkhorst and J. D. Pack, “Special points for Brillouin-zone integrations,” Physical Review B 13, 5188–5192 (1976), doi:10.1103/PhysRevB.13.5188.
• W. Setyawan and S. Curtarolo, “High-throughput electronic band structure calculations: Challenges and tools,” Computational Materials Science 49, 299–312 (2010), doi:10.1016/j.commatsci.2010.05.010.