Maximally-localized Wannier functions

Wannier functions are the localized, real-space dual of Bloch states: a unitary repackaging of an isolated group of bands into orbitals $|w_{n\mathbf{R}}⟩$ centered on lattice sites. The Bloch representation carries a gauge freedom — the bands are defined only up to a $\mathbf{k}$-dependent unitary mixing — and fixing that gauge by demanding maximal localization (Marzari–Vanderbilt) yields the MLWFs. Their two payoffs are (i) an exact, exponentially convergent real-space tight-binding Hamiltonian for Wannier interpolation, and (ii) localized orbitals that are the natural substrate for electron–phonon coupling and for the Green’s-function exchange branch.

Wannier functions and the gauge freedom

For a single isolated band, the Wannier function in cell $\mathbf{R}$ is the BZ Fourier transform of the Bloch state,

$$|w_{n\mathbf{R}}⟩=\frac{V}{(2\pi {)}^3}{\int }_{\mathrm{BZ}}d\mathbf{k}{e}^{-i\mathbf{k}\cdot \mathbf{R}}|{\psi }_{n\mathbf{k}}⟩.$$

The construction is far from unique. Bloch eigenstates of an isolated manifold of $J$ bands are fixed only up to a $\mathbf{k}$-dependent $J\times J$ unitary — the gauge — mixing them at each $\mathbf{k}$:

$$|{\tilde{\psi }}_{n\mathbf{k}}⟩=\sum _{m=1}^J{U}_{mn}^{(\mathbf{k})}|{\psi }_{m\mathbf{k}}⟩,U^{(\mathbf{k})}\in U(J).$$

For $J=1$ this is the familiar phase freedom $e^{i\phi (\mathbf{k})}$. The transformed Wannier functions

$$|w_{n\mathbf{R}}⟩=\frac{V}{(2\pi {)}^3}{\int }_{\mathrm{BZ}}d\mathbf{k}{e}^{-i\mathbf{k}\cdot \mathbf{R}}\sum _m{U}_{mn}^{(\mathbf{k})}|{\psi }_{m\mathbf{k}}⟩$$

depend strongly on the choice of $U^{(\mathbf{k})}$: their spatial spread and even their decay (exponential vs. power-law) hinge on the smoothness of the gauge across the BZ. A discontinuous or topologically obstructed gauge gives delocalized, ragged Wannier functions; the localization problem is therefore a search for a maximally smooth gauge. The phase convention here is the “periodic” gauge of Bloch’s theorem; the alternative atomic-gauge convention shifts Wannier centers by lattice vectors and must be tracked.

The spread functional (Marzari–Vanderbilt)

MLWFs minimize the total quadratic spread of the orbitals in the home cell,

$$\Omega =\sum _n[⟨w_{n0}|r^2|w_{n0}⟩-|⟨w_{n0}|\mathbf{r}|w_{n0}⟩{|}^2]=\sum _n(⟨r^2{⟩}_n-{\bar{\mathbf{r}}}_n^2),$$

with the Wannier center ${\bar{\mathbf{r}}}_n=⟨w_{n0}|\mathbf{r}|w_{n0}⟩$. Marzari and Vanderbilt’s key decomposition splits this into a gauge-invariant part and a gauge-dependent remainder,

$$\boxed{\Omega ={\Omega }_I+\tilde{\Omega }},{\Omega }_I=\sum _n[⟨r^2{⟩}_n-\sum _{m\mathbf{R}}|⟨w_{m\mathbf{R}}|\mathbf{r}|w_{n0}⟩{|}^2],\tilde{\Omega }=\sum _n\sum _{(m\mathbf{R})\ne (n0)}|⟨w_{m\mathbf{R}}|\mathbf{r}|w_{n0}⟩{|}^2.$$

${\Omega }_I$ is independent of $U^{(\mathbf{k})}$ — it is set by the band manifold itself (geometrically, the quantum-metric integral ${\Omega }_I=\frac{V}{(2\pi {)}^3}{\int }_{\mathrm{BZ}}\mathrm{tr}g(\mathbf{k})d\mathbf{k}$, a lower bound on the achievable spread). Only $\tilde{\Omega }\ge 0$ depends on the gauge, so minimizing $\Omega$ is minimizing $\tilde{\Omega }$. In practice $\Omega$ is evaluated on a uniform $\mathbf{k}$-mesh from the overlap matrices $M_{mn}^{(\mathbf{k},\mathbf{b})}=⟨u_{m\mathbf{k}}|u_{n,\mathbf{k}+\mathbf{b}}⟩$ between neighboring mesh points (finite-difference ${\nabla }_{\mathbf{k}}$), and the gauge is rotated by steepest-descent / conjugate-gradient on $U^{(\mathbf{k})}$ from a trial-orbital projection start (the wannier90 workflow). The minimizing $U^{(\mathbf{k})}$ generically exists and gives real, exponentially localized orbitals when the manifold is topologically trivial (no obstruction); a nonzero Chern number obstructs a smooth real gauge and the procedure cannot fully localize — the localization failure is a topological diagnostic in its own right.

Disentanglement (entangled / metallic bands)

The clean construction assumes an isolated manifold separated by gaps. For metals, or for $d$/$f$ bands hybridized into a wider window, the bands of interest are entangled with others over part of the BZ and a fixed $J$-band manifold does not exist globally. Souza–Marzari–Vanderbilt add an outer step: inside an energy window containing $N(\mathbf{k})\ge J$ Bloch states, choose at each $\mathbf{k}$ the optimal $J$-dimensional subspace — a rectangular isometry $U_{\mathbf{k}}^{\mathrm{dis}}\in {\mathbb{C}}^{N(\mathbf{k})\times J}$, $U_{\mathbf{k}}^{\mathrm{dis}†}{U}_{\mathbf{k}}^{\mathrm{dis}}=1_J$ — that minimizes the gauge-invariant spread ${\Omega }_I$, i.e. that varies as smoothly as possible from $\mathbf{k}$ to $\mathbf{k}$. This disentanglement is solved self-consistently as a subspace-selection problem (${\Omega }_I$ is the only piece sensitive to the choice of subspace), optionally with a frozen inner window in which selected states (e.g. bands straddling $E_F$) are kept exactly. The MLWF Wannier rotation $U^{(\mathbf{k})}\in U(J)$ then acts within the disentangled subspace. The output is a smooth $J$-band Bloch manifold that reproduces the DFT bands exactly inside the frozen window and optimally elsewhere.

Wannier interpolation

The principal use. Because MLWFs are exponentially localized, the Hamiltonian in the Wannier basis,

$$H_{mn}^{\mathrm{W}}(\mathbf{R})=⟨w_{m0}|H|w_{n\mathbf{R}}⟩,$$

decays rapidly in $\mathbf{R}$ and so is captured by a small set of $\{\mathbf{R}\}$ obtained from a coarse ab-initio $\mathbf{k}$-mesh. This is exactly a tight-binding real-space Hamiltonian, but first-principles-exact within the manifold rather than parametrized: an ordinary $J\times J$ eigenproblem (orthonormal Wannier basis, $S=1$)

$$H^{\mathrm{W}}(\mathbf{k})=\sum _{\mathbf{R}}{e}^{i\mathbf{k}\cdot \mathbf{R}}{H}^{\mathrm{W}}(\mathbf{R}),H^{\mathrm{W}}(\mathbf{k})c=E(\mathbf{k})c,$$

diagonalized at arbitrary $\mathbf{k}$ for negligible cost. This yields band energies, velocities (via ${\nabla }_{\mathbf{k}}{H}^{\mathrm{W}}$), Berry curvature, and DOS on ultradense meshes — convergence otherwise prohibitive with a full self-consistent calculation per $\mathbf{k}$. Wannier interpolation is the engine of EPW-style electron–phonon-coupling interpolation: the EPC matrix elements $g_{mn\nu }(\mathbf{k},\mathbf{q})$, computed on coarse $\mathbf{k}$ and $\mathbf{q}$ grids, are transformed into a localized Wannier–lattice representation $g_{mn\nu }({\mathbf{R}}_e,{\mathbf{R}}_p)$ (short-ranged in both electron and phonon cells), then interpolated to fine grids — the only tractable route to converged Eliashberg functions, mobilities, and zero-point band-gap renormalization, and the reason MLWFs sit upstream of the whole electron–phonon program. The same localized orbitals underpin the LKAG route to magnetic exchange (TB2J): a Wannier tight-binding model gives the intersite Green’s functions and on-site exchange splittings that the magnetic force theorem integrates into $J_{ij}$, so the exchange tensors and the EPC interpolation share this Wannier substrate. The cell-periodic gauge subtlety from the Bloch note resurfaces operationally here: the convention for the position phase fixes the Wannier centers and hence the long-range part of $g$ that must be treated analytically for polar materials.

Prerequisites

tight-binding & Bloch’s theorem · crystal symmetry

Builds toward: Green’s functions & LKAG · modern theory of polarization · electron–phonon-coupling interpolation

Key references

• Wannier functions & MLWF — G. H. Wannier, Phys. Rev. 52, 191 (1937); N. Marzari & D. Vanderbilt, PRB 56, 12847 (1997); review: N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza & D. Vanderbilt, Rev. Mod. Phys. 84, 1419 (2012).
• Disentanglement (entangled bands) — I. Souza, N. Marzari & D. Vanderbilt, PRB 65, 035109 (2001).
• Exponential localization / topological obstruction — C. Brouder, G. Panati, M. Calandra, C. Mourougane & N. Marzari, PRL 98, 046402 (2007).
• Electron–phonon Wannier interpolation (EPW) — F. Giustino, M. L. Cohen & S. G. Louie, PRB 76, 165108 (2007); review: F. Giustino, Rev. Mod. Phys. 89, 015003 (2017).
• Magnetic exchange (LKAG / TB2J) — A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov & V. A. Gubanov, JMMM 67, 65 (1987); X. He, N. Helbig, M. J. Verstraete & E. Bousquet, Comput. Phys. Commun. 264, 107938 (2021).