Wannier Functions

Wannier functions are lattice-translated real-space basis states constructed from a chosen Bloch-band subspace; a sufficiently regular, globally sewn frame makes them localized. They do not introduce new physical states: when the transform is exact, they reorganize the same retained subspace from crystal-momentum sectors into real-space orbitals. What makes the construction nontrivial is that Bloch eigenvectors are not unique. Their momentum-dependent phases, or unitary rotations within a composite band family, control whether the resulting real-space states are localized, where their centers lie, and which symmetries they display.

The canonical question is therefore not merely “Can I Fourier transform the bands?” On any finite momentum mesh, the answer is yes. The useful questions are sharper:

• Which rank-$J$ projector $P(\mathbf k)$ is being retained?
• Is it separated from its complement at every $\mathbf k$?
• Can one choose a globally sewn frame with the regularity needed for the requested localization?
• Must that frame also respect time reversal, a crystalline symmetry, or another constraint?
• Which later approximations—disentanglement, finite sampling, hopping truncation, or interaction projection—have been added to the exact basis change?

This page owns that construction and audit. Tight-Binding Models owns Hamiltonian modeling in a chosen localized basis; Chern Numbers in Band Theory and Topological Insulators own invariant and response physics; Hubbard Physics in Materials owns screened interactions and correlated-model validation.

Required background. Use the finite crystal-momentum mesh and translation convention from Bloch’s Theorem, the reciprocal torus and integration domain from Brillouin Zones, and orthogonal ranges and projector algebra from Projectors.

Helpful background. Fourier Series, Unitary Operators, Berry Connection, and the normalization and gauge ledger in Quantum Matter Conventions make the composite-band and center formulas easier to audit.

The Selected Bloch Subspace

Let $H(\mathbf k)$ be a one-particle Bloch Hamiltonian. Suppose a group of $J$ bands is separated from every excluded band by a direct gap at each momentum. If $\{|\psi_{m\mathbf k}\rangle\}_{m=1}^{J}$ is any orthonormal eigenframe for that group, its spectral projector is

$$P(\mathbf k) = \sum_{m=1}^{J} |\psi_{m\mathbf k}\rangle \langle\psi_{m\mathbf k}|.$$

The direct separation from the complement is the important condition. Bands inside the selected group may cross or be exactly degenerate. At such a crossing, energy-sorted eigenvectors can exchange labels or become numerically singular even while $P(\mathbf k)$ remains smooth. This is why composite-band Wannier theory begins with a projector rather than with a promise to track each energy branch separately.

An isolated family need not be occupied, and it need not be separated by an indirect insulating gap. Occupation and chemical potential matter for polarization and transport claims, but they are not required to ask whether a specified projector admits a localized basis.

At fixed $\mathbf k$, every other orthonormal frame spanning the same subspace has the form

$$\begin{aligned} |\widetilde\psi_{n\mathbf k}\rangle &= \sum_{m=1}^{J} |\psi_{m\mathbf k}\rangle U_{mn}(\mathbf k), \\ U(\mathbf k) &\in U(J). \end{aligned}$$

The projector is unchanged:

$$\sum_{n=1}^{J} |\widetilde\psi_{n\mathbf k}\rangle \langle\widetilde\psi_{n\mathbf k}| = P(\mathbf k).$$

For $J=1$, $U(\mathbf k)=e^{i\phi(\mathbf k)}$ is a phase. For $J>1$, the freedom is a full momentum-dependent unitary rotation. Wannier localization is primarily the problem of choosing this frame coherently across the Brillouin-zone torus.

Before doing so, declare:

1. the selected projector, its rank, and its separation from excluded states;
2. the finite-cell or infinite-crystal normalization convention;
3. the reciprocal-boundary sewing used for the frame;
4. whether ordinary exponential localization, compact support, or only hybrid localization is requested;
5. which symmetries the Wannier basis must represent explicitly;
6. the operator or observable for which the basis will be used; and
7. which steps are exact changes of basis and which are reductions or truncations.

The Finite Bloch–Wannier Transform

Start with a Born–von Karman crystal containing $N_c$ primitive cells. Its allowed momenta form a discrete set of $N_c$ points in one reciprocal primitive cell. Normalize the frame states by

$$\langle\widetilde\psi_{n\mathbf k} |\widetilde\psi_{m\mathbf k'}\rangle = \delta_{nm}\delta_{\mathbf k\mathbf k'}.$$

For every cell vector $\mathbf R$, define

$$|w_{n\mathbf R}\rangle = \frac{1}{\sqrt{N_c}} \sum_{\mathbf k} e^{-i\mathbf k\cdot\mathbf R} |\widetilde\psi_{n\mathbf k}\rangle.$$

The inverse transform is

$$|\widetilde\psi_{n\mathbf k}\rangle = \frac{1}{\sqrt{N_c}} \sum_{\mathbf R} e^{+i\mathbf k\cdot\mathbf R} |w_{n\mathbf R}\rangle.$$

These equations are an exact unitary discrete Fourier transform within the chosen subspace. They create $J N_c$ Wannier states from $J N_c$ Bloch states—no state is added or lost.

Translation covariance

The site convention is

$$T_{\mathbf A}|\widetilde\psi_{n\mathbf k}\rangle = e^{-i\mathbf k\cdot\mathbf A} |\widetilde\psi_{n\mathbf k}\rangle.$$

Inserting this eigenvalue into the Fourier sum shifts its cell label:

$$T_{\mathbf A}|w_{n\mathbf R}\rangle = |w_{n,\mathbf R+\mathbf A}\rangle.$$

One home-cell orbital thus generates the complete lattice family by translation.

Orthonormality

Using discrete Fourier orthogonality,

$$\frac{1}{N_c} \sum_{\mathbf k} e^{i\mathbf k\cdot(\mathbf R-\mathbf R')} = \delta_{\mathbf R\mathbf R'},$$

gives

$$\langle w_{n\mathbf R}|w_{m\mathbf R'}\rangle = \delta_{nm}\delta_{\mathbf R\mathbf R'}.$$

Standard Wannier functions are therefore orthonormal translations. Localized atomic trial functions, nonorthogonal orbitals, and projected trial states are useful ingredients in model building, but they are not automatically Wannier functions.

Completeness in the retained subspace

Summing over every translated orbital gives

$$\sum_{n,\mathbf R} |w_{n\mathbf R}\rangle \langle w_{n\mathbf R}| = \sum_{\mathbf k}P(\mathbf k).$$

The right-hand side is the projector onto the full selected finite-crystal subspace. Completeness is not completeness in the original continuum Hilbert space unless every band has been retained.

Infinite-crystal position representation

The finite formula is safest because its normalization is explicit. To connect it to the common integral expression, use the site’s cell-average convention

$$\langle u|v\rangle_c = \frac{1}{\Omega_c} \int_{\mathrm{cell}}d^dr\, u^\dagger(\mathbf r)v(\mathbf r),$$

with $\langle\widetilde u_{n\mathbf k}|\widetilde u_{m\mathbf k}\rangle_c=\delta_{nm}$. Since

$$\frac{1}{N_c}\sum_{\mathbf k} \longrightarrow \frac{\Omega_c}{(2\pi)^d} \int_{\mathrm{BZ}}d^dk,$$

the normalized thermodynamic-limit function is

\begin{aligned} w_{n\mathbf R}(\mathbf r) &= \frac{\sqrt{\Omega_c}}{(2\pi)^d} \int_{\mathrm{BZ}}d^dk \\[-0.2rem] &\quad\times e^{i\mathbf k\cdot(\mathbf r-\mathbf R)} \widetilde u_{n\mathbf k}(\mathbf r). \end{aligned}

The frequently seen coefficient $\Omega_c/(2\pi)^d$ instead corresponds to a unit cell-integral normalization. Mixing these conventions changes norms and matrix elements.

Gauge Freedom and Reciprocal Sewing

The frame rotation $U(\mathbf k)$ is not an electromagnetic gauge transformation. It is a basis choice inside $\operatorname{Ran}P(\mathbf k)$. It is also distinct from changing the embedding convention of an orbital basis. All three choices can alter intermediate connections or phases, so they must be recorded separately.

Momentum space is a torus, not an ordinary box. Opposite faces of a chosen Brillouin zone represent the same translation character, but cell-periodic basis vectors may be related there by a reciprocal sewing matrix. A valid global frame must obey that sewing convention. Demanding naïve componentwise periodicity in a basis whose components themselves are sewn can manufacture a false discontinuity.

A lattice-vector gauge shift

For one band, choose a new frame

$$|\widetilde\psi'_{\mathbf k}\rangle = e^{-i\mathbf k\cdot\mathbf R_0} |\widetilde\psi_{\mathbf k}\rangle,$$

where $\mathbf R_0$ is a lattice vector. Then

$$|w'_{\mathbf R}\rangle = |w_{\mathbf R+\mathbf R_0}\rangle.$$

The physical subspace is unchanged; only the labeling of its translated basis has shifted. A more general sewn periodic phase with zero winding can reshape the Wannier function without shifting its center by a lattice vector. For a composite family, $U(J)$ rotations can redistribute weight and centers among orbitals while preserving the total projector.

This is why onsite energies, individual centers, individual spreads, hopping amplitudes, and projected interaction tensors are generally basis dependent. Gauge-invariant observables are recovered only after all transformed operators and states are treated consistently.

Centers, Spreads, and Localization

Suppose a home-cell Wannier state has the required position moments. Its center and quadratic spread are

$$\overline{\mathbf r}_n = \langle w_{n\mathbf 0}|\mathbf r|w_{n\mathbf 0}\rangle,$$ $$\Omega = \sum_n \left( \langle r^2\rangle_n - |\overline{\mathbf r}_n|^2 \right).$$

Define the non-Abelian Berry connection of the chosen frame by

$$\mathbf A_{mn}(\mathbf k) = i\langle\widetilde u_{m\mathbf k}| \nabla_{\mathbf k} \widetilde u_{n\mathbf k}\rangle_c.$$

Then the center is

\begin{aligned} \overline{\mathbf r}_n &= \frac{\Omega_c}{(2\pi)^d} \int_{\mathrm{BZ}}d^dk\, \mathbf A_{nn}(\mathbf k) \\ &\hspace{1.5rem} \text{modulo a lattice vector}. \end{aligned}

The total spread can be decomposed schematically as

$$\Omega = \Omega_{\mathrm I} + \widetilde\Omega.$$

The invariant part $\Omega_{\mathrm I}$ depends only on the selected projector; $\widetilde\Omega$ depends on the chosen frame. Minimally spread Wannier functions optimize the latter within a declared subspace. This optimization selects a convenient representation, not a new physical ground state.

Wannier centers are not absolute electric dipoles. A polarization statement additionally needs the electron charge sign, filling, ionic contribution, origin, and branch convention. Berry-Phase Polarization and Charge Pumping owns that material ledger; this page retains localized-basis existence, gauge choice, and obstruction.

Regularity controls decay

Real-space localization follows from momentum-space regularity:

• finite differentiability supports corresponding algebraic decay or finite moments;
• a smooth frame can give decay faster than any fixed power under the usual regularity assumptions;
• exponential localization requires a frame analytic in a suitable complex-$\mathbf k$ neighborhood, not merely a visually smooth gauge on real momenta.

Compact support is stronger still and is not generic. Conversely, a finite $\mathbf k$ mesh always admits a discrete Fourier transform. A localized-looking finite-mesh orbital does not prove that a smooth, globally sewn thermodynamic family—or an exponentially localized limit—exists. Mesh refinement, real-space tails, spread convergence, and symmetry sewing must be checked.

Hamiltonians and Other Operators in the Wannier Basis

Let $\varepsilon(\mathbf k)$ be the diagonal matrix of retained Bloch energies in the original eigenframe. In the Wannier frame,

$$h^{\mathrm W}(\mathbf k) = U^\dagger(\mathbf k) \varepsilon(\mathbf k) U(\mathbf k).$$

With the hopping orientation used throughout this volume,

$$\begin{aligned} t_{mn}(\boldsymbol\Delta) &= \langle w_{m\mathbf 0}|H|w_{n\boldsymbol\Delta}\rangle \\ &= \frac{1}{N_c} \sum_{\mathbf k} e^{-i\mathbf k\cdot\boldsymbol\Delta} h^{\mathrm W}_{mn}(\mathbf k), \end{aligned}$$

and

$$h^{\mathrm W}_{mn}(\mathbf k) = \sum_{\boldsymbol\Delta} e^{+i\mathbf k\cdot\boldsymbol\Delta} t_{mn}(\boldsymbol\Delta).$$

Hermiticity gives

$$t_{mn}(\boldsymbol\Delta) = t_{nm}^{\ast}(-\boldsymbol\Delta).$$

These relations are exact inside the isolated selected subspace before any hopping is discarded. The same single-$\mathbf k$ rule applies to a lattice-translation-invariant projected one-body operator that is diagonal in crystal momentum: rotate its Bloch-space matrix by $U(\mathbf k)$ and Fourier transform it with the same convention. Position and dipole operators require derivative or connection terms, a general inhomogeneous one-body operator has a two-momentum kernel, and interactions require four orbital rotations and multi-momentum transforms. Transforming only the Hamiltonian is not enough to preserve every observable.

If hoppings outside a retained set $\mathcal S$ are omitted, then

$$\delta h(\mathbf k) = \sum_{\boldsymbol\Delta\notin\mathcal S} e^{i\mathbf k\cdot\boldsymbol\Delta} t(\boldsymbol\Delta),$$

so a conservative operator-norm bound is

$$\|\delta h(\mathbf k)\| \leq \sum_{\boldsymbol\Delta\notin\mathcal S} \|t(\boldsymbol\Delta)\|.$$

For a sufficiently local or regular Hamiltonian, localized Wannier functions often yield rapidly decaying hoppings, but that decay must still be checked. Range truncation is a modeling choice, not part of the definition of a Wannier basis.

Separate the error sources

The phrase “Wannier model” can hide several logically different steps:

1. Reference-Hamiltonian error: the original one-particle or mean-field Hamiltonian may already approximate the material.
2. Subspace projection: an isolated band family is retained and its complement is eliminated for a declared energy and observable window.
3. Disentanglement: if target bands overlap excluded bands, a smooth rank-$J$ subspace is selected from a larger energy window. This is not merely a gauge rotation of a fixed spectral projector.
4. Finite sampling: a discrete mesh approximates continuous Brillouin-zone information.
5. Real-space truncation: long-ranged matrix elements are dropped.
6. Interaction projection: Coulomb and other operators are mapped into the localized basis, with screening and double-counting choices.
7. Solver and observable error: the resulting model is solved and compared with a probe-specific quantity.

Only the frame rotation and complete Fourier transform within a fixed selected projector are exact basis changes. From Quantum Mechanics to Materials owns the end-to-end provenance ledger, while Computational Many-Body QM owns general convergence and numerical-evidence discipline.

Topological and Symmetry-Respecting Obstructions

The strongest existence statement must name both the selected subspace and the requested class of basis. Under the standard regularity assumptions in dimensions up to three, an analytic globally sewn Bloch frame yields exponentially localized, orthonormal, lattice-translated Wannier functions. Conversely, such a Wannier basis supplies a global analytic frame. The topology of the complex Bloch bundle can forbid that frame.

Chern obstruction

In two dimensions, a selected subspace with nonzero first Chern number cannot possess exponentially localized orthonormal Wannier functions spanning precisely that subspace. In three dimensions, the corresponding first-Chern numbers on the independent Brillouin-zone two-tori must vanish. This is a property of the selected projector, not of the Hamiltonian’s mere possession of some topological band.

The obstruction does not forbid every useful real-space construction. Hybrid Wannier functions can remain localized in one direction and extended in another. Algebraically localized or nonorthogonal functions may also be used. None of these is the requested fully exponentially localized orthonormal translated basis.

Nonzero local Berry curvature is not by itself an obstruction: its integral may vanish. A nonzero one-dimensional Zak phase is also not an obstruction. It can place a Wannier center at a nontrivial symmetry-allowed position while exponential localization remains possible.

Time-reversal and other symmetry constraints

A two-dimensional $\mathbb Z_2$-odd time-reversal insulator has zero total charge Chern number. It can admit exponentially localized Wannier functions if the frame is allowed to break the requirement that individual Wannier functions form time-reversal pairs. The obstruction is to a time-reversal-respecting gauge of that stronger kind, not to all exponentially localized bases.

Crystalline symmetries introduce further representation constraints. A symmetry-compatible basis may be obstructed even when the complex bundle is topologically trivial. A fragile obstruction can disappear after specified trivial bands are added. A stable Chern obstruction cannot. An obstructed atomic limit can already possess symmetric exponentially localized Wannier functions, but their centers occupy different Wyckoff positions from a chosen ionic reference. “Obstructed” therefore does not always mean “no Wannier functions.”

The symmetry labels, little-group representations, compatibility relations, and band-representation diagnostics belong to Symmetry of Bloch States. Here the stopping rule is simpler: state whether the requested basis is merely localized, or localized and constrained to furnish a symmetry-compatible Wannier representation at declared Wyckoff positions and site-symmetry representations, including any required local action of internal symmetries. Then route the remaining invariant or representation test to its canonical owner.

Worked Cases

A translated gauge in one dimension

For a one-dimensional lattice with $N_c$ cells, take a single isolated band and the gauge change

$$|\psi'_k\rangle = e^{-ikR_0}|\psi_k\rangle.$$

Substitution into the finite transform gives $|w'_R\rangle=|w_{R+R_0}\rangle$. Energies, the projector, and all exactly transformed observables are unchanged. The home-cell label and center branch shift by $R_0$. This example separates gauge-dependent orbital labels from gauge-invariant physics without invoking any approximation.

The SSH occupied band

In either gapped dimerization of the SSH Model, the isolated occupied band admits exponentially localized Wannier functions. With the symmetry and conventions declared, its center lies at one of the two symmetry-allowed inversion centers—commonly the intracell or intercell strong-bond center in the standard embedded chain. A site-centered interpretation requires a different or collapsed embedding. The distinction is meaningful only after specifying unit cell, origin, orbital embedding, termination for an edge claim, and ionic charges for a polarization claim.

A nonzero Zak phase in one convention does not imply a Chern obstruction. It records a center branch and, with symmetry, can diagnose an obstructed atomic limit. The bulk winding uses chiral symmetry and a declared unit-cell convention; the presence of a boundary state additionally depends on how the chain terminates.

One Chern band versus the complete orbital space

Consider a two-orbital Chern-insulator Hamiltonian. If the lower band has $C=1$, that rank-one projector has no exponentially localized orthonormal Wannier basis. Select both bands instead. The complete two-orbital Hilbert bundle has the original $\mathbf k$-independent cell-orbital frame and is topologically trivial; compact localized cell orbitals already span it.

There is no contradiction. The obstruction belonged to the rank-one selected projector, not to the full Hamiltonian. Chern Numbers in Band Theory owns the invariant calculation and Hall-response consequences.

A composite subspace through an internal crossing

Suppose two retained bands cross each other while remaining directly separated from all excluded bands. Tracking the lower and upper eigenvector by energy can swap their character at the crossing. The rank-two projector remains regular, and a smooth two-component frame may pass through the crossing without singularity.

This is the practical reason to Wannierize a composite subspace rather than insist that each sorted energy branch define an independent orbital. The frame may mix the two eigenvectors, so $h^{\mathrm W}(\mathbf k)$ need not be diagonal even though its eigenvalues reproduce both retained bands exactly.

Scope and Handoffs

This page stops after the exact theory and its validity ledger.

• Tight-Binding Models owns hopping-range choices, orbital embeddings, spin–orbit terms, material fitting, and validation of the resulting Bloch Hamiltonian.
• Topology in Quantum Matter owns the taxonomy of band topology, symmetry-protected phases, and intrinsic topological order.
• Flat Bands owns compact-localization constraints and interaction criteria for nearly flat bands.
• Hubbard Physics in Materials owns active-space selection, screened interaction tensors, double counting, and material-spectral validation.
• Band Structure Workflows owns the converged source calculation, full-zone isolation checks, symmetry record, and band/subspace artifacts that precede a numerical localized-orbital construction.
• Wannierization Workflows owns material-facing numerical projections, outer and frozen energy windows, disentanglement, spread minimization, interpolation files, validation, and reproducibility records; this page retains the exact existence-and-representation theory.

Common Pitfalls

Treating every Fourier-transformed finite mesh as localized Wannier theory

A finite mesh always yields discrete transformed vectors. Exponential localization and topological obstruction are thermodynamic regularity statements.

Tracking energy labels instead of the projector

Internal degeneracies can make individual eigenvectors discontinuous while the retained composite subspace remains smooth.

Equating a smooth plot with exponential localization

Real-axis smoothness is not the analytic continuation condition required for exponential decay.

Calling any localized orbital a Wannier function

Atomic trial functions and nonorthogonal orbitals need not be orthonormal translates spanning an exact Bloch subspace.

Treating Wannier centers or hoppings as observables

They depend on frame, embedding, origin, and branch. Only consistently reconstructed physical predictions are basis independent.

Saying that all topological bands forbid Wannier functions

Chern, time-reversal-respecting, fragile, and obstructed-atomic-limit statements impose different conditions and have different remedies.

Forgetting the scope of ordinary translations

Magnetic translations, disorder, open boundaries, and incommensurate structures require modified constructions. This page concerns ordinary lattice translations in one-particle or mean-field Bloch subspaces; it does not assign one-particle Wannier functions to generic interacting many-body eigenstates.

Exercises

1. Prove the finite transform identities

Starting from the finite Bloch–Wannier transform, derive translation covariance, orthonormality, the inverse transform, and completeness in the retained subspace.

Solution

Insert the translation eigenvalue $e^{-i\mathbf k\cdot\mathbf A}$ into the forward transform to obtain $T_{\mathbf A}|w_{n\mathbf R}\rangle=|w_{n,\mathbf R+\mathbf A}\rangle$. For the overlap, orthonormality of the Bloch frame reduces the double momentum sum to

$$\frac{\delta_{nm}}{N_c} \sum_{\mathbf k} e^{i\mathbf k\cdot(\mathbf R-\mathbf R')} = \delta_{nm}\delta_{\mathbf R\mathbf R'}.$$

Multiplying the forward transform by $e^{i\mathbf k\cdot\mathbf R}/\sqrt{N_c}$ and summing over $\mathbf R$ gives the inverse. Substituting either pair into $\sum_{n\mathbf R}|w_{n\mathbf R}\rangle\langle w_{n\mathbf R}|$ collapses the cell sum and yields $\sum_{\mathbf k}P(\mathbf k)$.

2. Shift a Wannier center by gauge

For one dimension, apply $e^{-ikR_0}$ to a single-band frame. Then apply a sewn periodic phase $e^{i\phi(k)}$ with zero winding. What changes in each case?

Solution

The first phase gives $w'_R=w_{R+R_0}$ and shifts the center branch by the lattice vector $R_0$. It does not change the projector or any consistently transformed observable. A zero-winding periodic phase can reshape the orbital and change its spread, but its Berry-connection integral changes by the integral of $-\partial_k\phi$, which vanishes over the Brillouin zone. The center modulo a lattice vector is unchanged.

3. Derive hopping Hermiticity and a truncation bound

Use $t_{mn}(\boldsymbol\Delta)=\langle w_{m0}|H|w_{n\boldsymbol\Delta}\rangle$ to prove the stated Hermiticity relation. Then bound the Bloch-Hamiltonian error after omitting a set of hoppings.

Solution

Hermiticity gives

$$t_{mn}(\boldsymbol\Delta) = \langle w_{n\boldsymbol\Delta}|H|w_{m0}\rangle^\ast.$$

Translate both states by $-\boldsymbol\Delta$ and use translation invariance to obtain $t_{mn}(\boldsymbol\Delta)=t_{nm}^\ast(-\boldsymbol\Delta)$. The omitted contribution is a sum of phase-weighted matrices. Since each phase has unit modulus, the triangle inequality gives $\|\delta h(\mathbf k)\|\leq\sum_{\mathrm{omitted}}\|t(\boldsymbol\Delta)\|$.

4. Diagnose a Chern obstruction

A numerically isolated band has $C=1$. A student requests exponentially localized orthonormal Wannier functions spanning only that band. Explain the obstruction, then explain why adding the complementary band of a complete two-orbital model can remove this particular obstruction.

Solution

An exponentially localized orthonormal translated basis would define a global analytic sewn Bloch frame. Such a frame makes the selected complex line bundle trivial and forces its first Chern number to vanish, contradicting $C=1$. The complete two-orbital space has the original global cell-orbital frame. Its total Chern number is zero, so selecting both bands removes the rank-one Chern obstruction, although other imposed symmetry constraints would still need a separate audit.

5. Rotate a composite-band frame

At one momentum, let $\varepsilon=\operatorname{diag}(\varepsilon_1,\varepsilon_2)$ and rotate its two eigenvectors with

$$U(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{pmatrix}.$$

Show that the rank-two projector is invariant. Compute $h^{\mathrm W}=U^\dagger\varepsilon U$, identify when it is off diagonal, and explain how a smooth composite frame can pass through an internal band crossing even when an energy-sorted eigenframe swaps labels. What can then happen to $U(\mathbf k)$ defined relative to that sorted frame?

Solution

Unitary rotation preserves the frame sum:

$$\sum_{n=1}^{2}|\widetilde\psi_n\rangle \langle\widetilde\psi_n| = \sum_{m=1}^{2}|\psi_m\rangle\langle\psi_m|.$$

Writing $c=\cos\theta$ and $s=\sin\theta$ gives

$$h^{\mathrm W} = \begin{pmatrix} a & b\\ b & d \end{pmatrix}.$$

Here

$$\begin{aligned} a&=\varepsilon_1c^2+\varepsilon_2s^2, \\ b&=(\varepsilon_2-\varepsilon_1)cs, \\ d&=\varepsilon_1s^2+\varepsilon_2c^2. \end{aligned}$$

It is generically off diagonal when the energies differ and the rotation mixes the eigenvectors. At the degeneracy itself, $\varepsilon_1=\varepsilon_2$ and the matrix is proportional to the identity, so every $U(2)$ choice gives the same $h^{\mathrm W}$ at that momentum; regularity in a neighborhood is a separate condition. One may relabel or rotate the degenerate eigenframe into a smooth reference frame. If one instead keeps a discontinuous energy-sorted reference, the compensating $U(\mathbf k)$ generally contains the inverse swap and can be discontinuous even though the product $|\widetilde\psi_n(\mathbf k)\rangle=\sum_m|\psi_m(\mathbf k)\rangle U_{mn}(\mathbf k)$ is smooth. The rank-two projector and reproduced eigenvalues remain unchanged.

6. Audit a finite-mesh claim

A calculation on an $8\times8$ mesh produces compact-looking discrete Fourier orbitals and declares the thermodynamic subspace exponentially Wannierizable. What evidence is missing?

Solution

Every finite mesh can be Fourier transformed, so the observation is not an existence test. If the claim concerns an isolated spectral band family, verify its direct separation from the complement. If it concerns a deliberately disentangled, non-spectral rank-$J$ projector, declare how that projector was selected and test its dependence on outer and inner windows and trial states. In either case, verify reciprocal sewing, mesh convergence, stable centers and spreads, decaying real-space tails, and the relevant Chern or symmetry obstruction tests.

7. Sort exact and approximate steps

Classify the following operations: a $U(J)$ frame rotation inside a fixed projector; the complete Bloch–Wannier transform; projection onto an isolated band family; disentanglement from overlapping bands; finite-$\mathbf k$ sampling; hopping truncation; and projection of Coulomb interactions.

Solution

The frame rotation and complete finite transform are exact changes of basis inside the fixed subspace. Isolated-band projection is exact as a definition of the retained subspace but is a physical reduction relative to the full Hilbert space, whose adequacy depends on the target energy and observable. Disentanglement selects a new smooth subspace and is model dependent. Finite sampling and hopping truncation are numerical and real-space approximations. Interaction projection is exact only if every induced matrix element and eliminated-sector effect is retained; practical screened low-energy models add approximation and matching choices.

References

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• W. Kohn, “Analytic Properties of Bloch Waves and Wannier Functions,” Physical Review 115, 809–821 (1959), doi:10.1103/PhysRev.115.809. The theorem proved there is scoped to a one-dimensional centrosymmetric setting.
• N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, “Maximally Localized Wannier Functions: Theory and Applications,” Reviews of Modern Physics 84, 1419–1475 (2012), doi:10.1103/RevModPhys.84.1419.
• N. Marzari and D. Vanderbilt, “Maximally Localized Generalized Wannier Functions for Composite Energy Bands,” Physical Review B 56, 12847–12865 (1997), doi:10.1103/PhysRevB.56.12847.
• G. Panati, “Triviality of Bloch and Bloch–Dirac Bundles,” Annales Henri Poincaré 8, 995–1011 (2007), doi:10.1007/s00023-007-0326-8.
• H. C. Po, H. Watanabe, and A. Vishwanath, “Fragile Topology and Wannier Obstructions,” Physical Review Letters 121, 126402 (2018), doi:10.1103/PhysRevLett.121.126402.
• A. A. Soluyanov and D. Vanderbilt, “Wannier Representation of $\mathbb Z_2$ Topological Insulators,” Physical Review B 83, 035108 (2011), doi:10.1103/PhysRevB.83.035108.
• I. Souza, N. Marzari, and D. Vanderbilt, “Maximally Localized Wannier Functions for Entangled Energy Bands,” Physical Review B 65, 035109 (2001), doi:10.1103/PhysRevB.65.035109.
• D. Vanderbilt, Berry Phases in Electronic Structure Theory, Cambridge University Press, 2018, doi:10.1017/9781316662205.
• G. H. Wannier, “The Structure of Electronic Excitation Levels in Insulating Crystals,” Physical Review 52, 191–197 (1937), doi:10.1103/PhysRev.52.191.