3.2 Wannier functions

Definition and concept

Wannier functions are localized basis functions that offer an alternative to the extended Bloch states you're used to from band theory. While Bloch states are delocalized across the entire crystal (labeled by crystal momentum $\mathbf{k}$), Wannier functions are centered on specific lattice sites and give you a real-space picture of the electronic structure. This makes them especially useful for understanding chemical bonding, local interactions, and any problem where thinking in real space is more natural than thinking in $\mathbf{k}$-space.

Localized basis functions

Wannier functions look somewhat like atomic orbitals, but they're not the same thing. They account for the full periodic potential of the crystal lattice, so they already "know" about the crystalline environment. Because they're localized, they let you talk about electronic properties at a particular site, which is exactly what you need for describing things like hopping between atoms or local Coulomb interactions.

Relation to Bloch functions

The key idea is that Wannier functions and Bloch functions contain exactly the same information. You can go from one to the other through a unitary transformation, meaning nothing is lost or gained. Bloch states spread over the whole crystal; Wannier functions are localized. You pick whichever representation makes your problem easier.

Fourier transform of Bloch states

Concretely, you obtain Wannier functions by Fourier transforming Bloch states over the Brillouin zone:

$w_n(\mathbf{r} - \mathbf{R}) = \frac{V}{(2\pi)^3} \int_{\text{BZ}} e^{-i\mathbf{k}\cdot\mathbf{R}} \psi_{n\mathbf{k}}(\mathbf{r})\, d\mathbf{k}$

Here $n$ is the band index, $\mathbf{R}$ is a lattice vector labeling which unit cell the function is centered on, and the integral runs over all $\mathbf{k}$ in the first Brillouin zone. The inverse transformation reconstructs the Bloch states from the Wannier functions:

$\psi_{n\mathbf{k}}(\mathbf{r}) = \sum_{\mathbf{R}} e^{i\mathbf{k}\cdot\mathbf{R}} w_n(\mathbf{r} - \mathbf{R})$

This pair of transforms is the formal bridge between reciprocal space and real space.

Mathematical formulation

Wannier function construction

For a finite crystal with $N$ unit cells, the Fourier transform becomes a discrete sum:

$w_n(\mathbf{r} - \mathbf{R}) = \frac{1}{\sqrt{N}} \sum_{\mathbf{k}} e^{-i\mathbf{k}\cdot\mathbf{R}} \psi_{n\mathbf{k}}(\mathbf{r})$

The sum runs over all $N$ allowed $\mathbf{k}$-points in the Brillouin zone. The result is a set of functions, one per unit cell, each centered at lattice site $\mathbf{R}$. The $1/\sqrt{N}$ prefactor ensures proper normalization. A critical subtlety is that the Bloch states $\psi_{n\mathbf{k}}$ must have smoothly varying phases as a function of $\mathbf{k}$; a careless choice of phases will produce poorly localized or even discontinuous Wannier functions.

Gauge freedom

This phase issue is the gauge freedom problem. For any band $n$, you can multiply the Bloch state by a $\mathbf{k}$-dependent phase factor:

$\psi_{n\mathbf{k}}(\mathbf{r}) \rightarrow e^{i\phi_n(\mathbf{k})} \psi_{n\mathbf{k}}(\mathbf{r})$

This transformation doesn't change any physical observable (eigenvalues, charge density, etc.), but it does change the resulting Wannier functions. Different gauge choices produce Wannier functions with different shapes and different degrees of localization. This is why finding maximally localized Wannier functions (the Marzari-Vanderbilt procedure) is such an important problem: you optimize the gauge choice to minimize the total spread $\Omega = \sum_n \left[\langle w_n | r^2 | w_n \rangle - \langle w_n | \mathbf{r} | w_n \rangle^2 \right]$.

For composite bands (multiple bands that are entangled or connected), the gauge freedom generalizes to a $\mathbf{k}$-dependent unitary mixing among bands:

$\psi_{n\mathbf{k}} \rightarrow \sum_m U_{mn}(\mathbf{k})\, \psi_{m\mathbf{k}}$

This broader freedom is what makes the construction of Wannier functions for realistic multi-band systems both powerful and nontrivial.

Periodic boundary conditions

There's an important distinction to keep straight here. Wannier functions are not periodic in $\mathbf{r}$. The correct statement is that the set of Wannier functions is closed under lattice translations: the Wannier function centered at $\mathbf{R}$ is just the one centered at the origin, shifted by $\mathbf{R}$:

$w_n(\mathbf{r} - \mathbf{R})$

This means all Wannier functions for a given band $n$ have the same shape, just translated to different lattice sites. The periodicity of the crystal enters through the Bloch states used in the construction and through the requirement that the gauge choice $\phi_n(\mathbf{k})$ (or $U_{mn}(\mathbf{k})$) be periodic in the reciprocal lattice.

Properties of Wannier functions

Spatial localization

Wannier functions decay with distance from their center. For isolated bands with a finite energy gap above and below, it can be proven that exponentially localized Wannier functions exist. The rate of exponential decay is related to the analyticity of the Bloch states as functions of $\mathbf{k}$, which in turn depends on the size of the band gap. Larger gaps generally allow tighter localization.

The degree of localization also depends on the gauge choice. A random gauge typically gives Wannier functions that decay as a power law, which is much worse. This is why the maximally localized construction matters in practice.

Orthogonality

Wannier functions form a complete orthonormal set:

$\langle w_n(\mathbf{r} - \mathbf{R}) | w_m(\mathbf{r} - \mathbf{R}') \rangle = \delta_{nm}\, \delta_{\mathbf{R}\mathbf{R}'}$

They're orthogonal both across different bands ($n \neq m$) and across different lattice sites ($\mathbf{R} \neq \mathbf{R}'$). This orthonormality follows directly from the orthonormality of Bloch states and the unitarity of the Fourier transform. It means the Wannier functions form a legitimate basis: you can expand any state in the relevant bands as a linear combination of Wannier functions, and matrix elements in this basis are straightforward to compute.

Symmetry considerations

Wannier functions can be constructed to respect the point group and space group symmetry of the crystal. When this is done, they transform according to irreducible representations of the site symmetry group at their center. For example, in a cubic crystal you might get Wannier functions with $s$-like, $p$-like, or $d$-like symmetry at each site.

Symmetry-adapted construction is not just aesthetically nice. It reduces the number of independent parameters in tight-binding models and provides a direct connection to the atomic orbital character of bands. It also plays a role in topological classification, since certain symmetry representations can obstruct the construction of symmetric, localized Wannier functions.

Applications in condensed matter

Tight-binding models

Wannier functions are the natural basis for tight-binding Hamiltonians. The Hamiltonian matrix elements in the Wannier basis are the hopping integrals:

$t_{nm}(\mathbf{R}) = \langle w_n(\mathbf{r}) | \hat{H} | w_m(\mathbf{r} - \mathbf{R}) \rangle$

Because Wannier functions are localized, these hopping integrals decay with $|\mathbf{R}|$, so you can truncate to near neighbors and still get an accurate band structure. This is exactly how tools like Wannier90 work: they take a first-principles DFT calculation, construct maximally localized Wannier functions, and output a tight-binding Hamiltonian that faithfully reproduces the ab initio bands. This tight-binding form is then cheap enough to use for large-scale transport calculations, Fermi surface interpolation, or as input to many-body methods like dynamical mean-field theory (DMFT).

Chemical bonding analysis

The centers and spreads of Wannier functions give direct information about bonding. In a covalent material, you'll find Wannier function centers sitting between atoms (along bonds). In an ionic material, they cluster around the anion. In metals, they tend to be more spread out.

This analysis connects to the modern theory of polarization: the electronic contribution to the macroscopic polarization of a crystal is determined by the sum of the Wannier function centers. Changes in polarization (as in a ferroelectric) correspond to shifts of these centers, giving a rigorous and gauge-invariant way to compute polarization from first principles.

Topological insulators

Topology and Wannier functions have a deep connection. For a topologically trivial insulator, you can always construct a set of exponentially localized Wannier functions that spans the occupied bands. For a topologically nontrivial insulator (like a $\mathbb{Z}_2$ topological insulator or a Chern insulator), this construction is obstructed: no smooth, periodic gauge choice exists across the entire Brillouin zone.

• For a 2D system with nonzero Chern number $C$, exponentially localized Wannier functions simply cannot be constructed for the occupied bands.
• Hybrid Wannier functions, localized in one direction but extended in another, can still be defined. Their centers as a function of the remaining momentum wind across the unit cell, and the winding number equals the Chern number.
• This framework provides a real-space perspective on bulk-boundary correspondence and is used to compute $\mathbb{Z}_2$ invariants and other topological indices.

The inability to construct well-localized Wannier functions is not a technical failure; it's a physical signature of nontrivial band topology.