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advanced27 min studyLesson 508 in Physics

Turn a band subspace into localized orbitals

Use unitary Fourier transforms while keeping gauge freedom and orthogonality explicit.

Before you begin
On this page

The question for today

What you’ll understand

  • Construct finite-crystal Wannier states with the correct normalization.
  • Explain how k-dependent phases change localization without changing band energies.
  • Distinguish localized trial orbitals from an orthonormal Wannier basis.

Bloch states display translation symmetry but are extended in space. Wannier states reorganize the same band subspace into orbitals centered around different cells. Nothing has been added to or removed from the Hilbert space when the transformation is unitary. Localization depends on coherent phases across wavevector, so a set of band energies alone cannot determine a unique Wannier orbital. This freedom is useful, but it comes with orthogonality and possible topological restrictions.

Fourier-transform an entire band

In a finite periodic crystal, choose normalized Bloch states for all N allowed wavevectors in an isolated band. Sum them with a cell-dependent negative Fourier phase and divide by the square root of N. Orthogonality of the Bloch states and the discrete Fourier identity then make the Wannier states orthonormal between cells. The inverse transform reconstructs every Bloch state, demonstrating that the number of states is unchanged and that localization is a change of basis.

For several bands spanning an isolated subspace, one may additionally mix their cell-periodic states with a wavevector-dependent unitary matrix before Fourier transforming. This does not preserve each individual energy eigenstate as a labeled orbital, but it preserves the whole selected subspace. Choosing a band window, or disentangling it from overlapping bands, is an additional modeling step. A localized basis constructed from an inadequate subspace can miss important states even if its orbitals look chemically appealing.

∣wnR⟩=1N∑ke−ik⋅R∣ψnk⟩,⟨wnR∣wmR′⟩=δnmδRR′|w_{n\mathbf R}\rangle=\frac1{\sqrt N}\sum_{\mathbf k}e^{-i\mathbf k\cdot\mathbf R}|\psi_{n\mathbf k}\rangle,\qquad \langle w_{n\mathbf R}|w_{m\mathbf R'}\rangle=\delta_{nm}\delta_{\mathbf R\mathbf R'}
N is the finite number of primitive cells and allowed k points, and Bloch states have unit norm in that crystal. The displayed formula uses a fixed orthonormal isolated-band gauge; composite-band unitary mixing is allowed before the transform.↔ Scroll wide equations horizontally.

Gauge affects shape and center

Multiplying each Bloch state by a different phase leaves its eigenvalue unchanged but changes interference among wavevectors in a Wannier sum. Smooth phases tend to support localization; rapidly varying phases can spread an orbital over distant cells. A phase linear in k shifts its center by a lattice vector, making the conventional labeling of a Wannier center nonunique by lattice translations. Energy bands therefore do not fix absolute orbital centers or a unique real-space spread.

Maximally localized Wannier methods choose a gauge by minimizing a spread functional built from position moments, often with additional symmetry or chemical goals. They do not remove every possible degeneracy or turn all localized orbitals into unique observables. In bands with nonzero Chern topology, a globally smooth periodic gauge supporting exponentially localized orthonormal Wannier functions for that isolated subspace is obstructed. The ordinary localized construction must therefore state its band-subspace and topological assumptions.

Ωspread=∑n(⟨wn0∣r2∣wn0⟩−∣⟨wn0∣r∣wn0⟩∣2)\Omega_{\rm spread}=\sum_n\left(\langle w_{n0}|r^2|w_{n0}\rangle-|\langle w_{n0}|\mathbf r|w_{n0}\rangle|^2\right)
The spread has units m². It is a gauge-dependent localization objective within a chosen subspace, distinct from the Berry curvature also often denoted Ω; finite second moments are assumed.↔ Scroll wide equations horizontally.

Localized does not automatically mean orthogonal

A translated set of narrow atomic-like functions can overlap strongly, even if each function is visually localized. Such a set is not automatically the orthonormal Wannier basis obtained from a unitary transform. Check overlaps as well as individual norms. Wannier Hamiltonian matrix elements can then be used for band interpolation, but truncating long-range terms introduces an approximation whose errors should be checked against the original band subspace, especially for derivatives and small avoided crossings.

The waves lab illustrates one coherent-cancellation step only. Reset and set phase difference to 180° and amplitude ratio to 1; predict resultant amplitude zero. Reset and set phase difference to 90° and amplitude ratio to 1; predict resultant amplitude square root of two. These equal-frequency waves show why phases affect sums even when component amplitudes agree. They are not a discrete Fourier transform over crystal momenta, a Wannier spread optimizer, or a demonstration of topological localization.

∣ψnk⟩↦e−ik⋅R0∣ψnk⟩⟹∣wnR′⟩=∣wn,R+R0⟩|\psi_{n\mathbf k}\rangle\mapsto e^{-i\mathbf k\cdot\mathbf R_0}|\psi_{n\mathbf k}\rangle\quad\Longrightarrow\quad |w'_{n\mathbf R}\rangle=|w_{n,\mathbf R+\mathbf R_0}\rangle
R₀ is a lattice translation in m. This sign follows the negative Fourier phase in our Wannier definition; changing conventions consistently leaves the translated physical basis equivalent.↔ Scroll wide equations horizontally.

Work it through

Four sites and coherent cancellation

On a four-site ring with orthonormal site states |j⟩, define |ψkm⟩=½Σj exp(ikmja)|j⟩ with km=2πm/(4a). Construct w₀=½Σm|ψkm⟩ and determine what happens if every Bloch state acquires exp(−ikma).

  1. The coefficient of site j in w₀ is one quarter of the sum of exp(i2πmj/4) over m=0,1,2,3. For j=0 all four contributions equal one, giving coefficient one after the quarter factor.

  2. For j=1 the four phases are 1,i,−1,−i and sum to zero. The other nonzero site indices likewise cancel by the discrete Fourier identity. Thus w₀ is exactly the site state |0⟩ in this complete one-orbital toy band.

  3. Adding phase exp(−ikma) changes the sum to exp[ikm(j−1)a]. It now selects j=1, so the transformed w₀ is |1⟩. Every Bloch energy is unchanged while the Wannier label’s center shifts by one cell.

The result

The original w₀ is |0⟩. The stated k-dependent phase produces w′₀=|1⟩ without changing any band energy.

The living notebook

Explore the model

Interactive lab

Interactive lab · waves

Test phase-sensitive addition

Reset and set phase difference=180°, amplitude ratio=1; predict resultant amplitude 0. Reset and set phase difference=90°, amplitude ratio=1; predict √2.

Test phase-sensitive additionTwo waves share the same frequency and wave number. Playback advances t in sin(2x − t) + a sin(2x − t + phase). Current readings: Sum at x = 0: 0.866; Resultant amplitude: 1.732; Phase difference: 60°. All parameters are adjustable with the labeled controls below.−3.14−1.5701.573.14−3.4−1.701.73.4x · radiansAmplitude
SumFirst componentSecond component
Sum at x = 0
0.866
Resultant amplitude
1.732
Phase difference
60°
t = 0 s

Two waves share the same frequency and wave number. Playback advances t in sin(2x − t) + a sin(2x − t + phase). Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe two fixed-frequency traveling sine waves illustrate coherent addition only. They do not Fourier-transform Bloch states or compute localized orbitals and their spreads.

Pause & reason

Check your understanding

What can change when Bloch states are multiplied by a k-dependent phase without changing their energies?

Your turn

Try it yourself

On an infinite orthonormal site chain, define trial orbitals w₀=(|0⟩+|1⟩)/√2 and w₁=(|1⟩+|2⟩)/√2. Find their overlap and w₀’s position center and variance for sites at ja. Can these translated trials be an orthonormal Wannier family?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Normalize each orbital separately, then keep the shared site in the cross overlap.

Compare with the solution

The shared site gives ⟨w₀|w₁⟩=1/2, so separate unit norms do not make the family orthogonal. For w₀, mean position is a/2 and second moment a²/2, giving variance a²/4. These localized translated trials are not the orthonormal Wannier family of a unitary isolated-band transform unless their overlap is addressed by a different construction.

Keep the idea

Take it with you

  • Wannier and Bloch states can span exactly the same band subspace.
  • Wavevector-dependent gauges change localization and center conventions.
  • Localization, orthogonality, and topology impose separate requirements.

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