Math 2.0

Statistical mechanics, structure of matter Definition

Wannier function

Builds on Electronic band structureBrillouin zoneTight-binding model

Definition

Let Λ⊂ℝn\Lambda \subset \R^n be a Bravais lattice with primitive cell Ω\Omega of volume ∣Ω∣\abs{\Omega}, reciprocal lattice Λ∗\Lambda^* and Brillouin zone BB. Let H=−Δ+VH = -\Delta + V be a periodic Schrödinger operator, and let EmE_m be a band of its band structure that does not meet the other bands. Choose Bloch functions ψk\psi_k of this band for k∈ℝnk \in \R^n, measurable in kk, such that:

  1. ψk(x+R)=eik⋅Rψk(x)\psi_k(x + R) = e^{ik \cdot R} \psi_k(x) for all R∈ΛR \in \Lambda, and Hψk=Em(k)ψkH\psi_k = E_m(k)\psi_k;
  2. ∫Ω∣ψk(x)∣2 dx=1\int_\Omega \abs{\psi_k(x)}^2 \, dx = 1;
  3. ψk+G=ψk\psi_{k + G} = \psi_k for all G∈Λ∗G \in \Lambda^*.

The Wannier function of the band at the lattice site R∈ΛR \in \Lambda is wR(x)=∣Ω∣(2π)n∫Be−ik⋅R ψk(x) dk.w_R(x) = \frac{\abs{\Omega}}{(2\pi)^n} \int_B e^{-ik \cdot R} \, \psi_k(x) \, dk.

Remarks

Clause 1 gives wR(x)=w0(x−R)w_R(x) = w_0(x - R), and the functions wRw_R, R∈ΛR \in \Lambda, form an orthonormal basis of the spectral subspace of the band. The Bloch functions are recovered as ψk(x)=∑R∈Λeik⋅RwR(x)\psi_k(x) = \sum_{R \in \Lambda} e^{ik \cdot R} w_R(x). The choice of phases ψk↦eiφ(k)ψk\psi_k \mapsto e^{i\varphi(k)}\psi_k changes the Wannier functions. If the phases depend smoothly on kk, then w0w_0 decays fast; the existence of exponentially decaying Wannier functions is obstructed by a nonzero Chern number of the band.

Example: for the one-band tight-binding model on ℓ2(ℤ)\ell^2(\Z), the lattice analogue has Ω={0}\Omega = \set{0} and ψk(n)=eikn\psi_k(n) = e^{ikn}, and wR(n)=12π∫−ππe−ikReikn dk=δnR,w_R(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{-ikR} e^{ikn} \, dk = \delta_{nR}, the function concentrated on the site RR.

Non-example: a Bloch function ψk\psi_k itself is not a Wannier function: it has ∣ψk(x+R)∣=∣ψk(x)∣\abs{\psi_k(x + R)} = \abs{\psi_k(x)} for all RR, so it is not square-integrable on ℝn\R^n.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Almost Mathieu operator, example, no evidence levelAlmost Mathieu operat…exampleBrillouin zone, concept, no evidence levelBrillouin zoneconceptElectronic band structure, concept, no evidence levelElectronic band struc…conceptSelf-adjoint operator, concept, no evidence levelSelf-adjoint operatorconceptTight-binding model, example, no evidence levelTight-binding modelexampleWannier function, definition, no evidence levelWannier functiondefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Almost Mathieu operator example
  2. Brillouin zone concept
  3. Electronic band structure concept
  4. Self-adjoint operator concept
  5. Tight-binding model example
  6. Wannier function definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Wannier_function
About this page
TagMT86BG43B0
URIm2://MT86BG43B0
Kinddefinition
WritingW0 stub
MSC82D20, 81Q10
DefinesWannier function
UsesElectronic band structure, Brillouin zone, Tight-binding model
Changed3 November 2026

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