Band-Resolved Wannier-Sector Fragmentation

• Band-resolved Wannier-sector fragmentation is the controlled decomposition of band manifolds into localized sectors distinguished by orbital content, symmetry, and topology.
• It leverages advanced disentanglement and localization techniques to isolate contributions from impurities, substrates, and surfaces within complex electronic systems.
• This approach is pivotal for analyzing optical responses, topological obstructions, and disorder effects by enabling a sector-resolved interpretation of band structures.

“Band-resolved Wannier-sector fragmentation” (Editor's term) denotes the decomposition of a band manifold into Wannier-defined sectors that are resolved by orbital character, symmetry action, topology, or transition channel. In the standard maximally-localized Wannier-function construction, Bloch eigenstates are transformed into localized orthonormal functions, enabling the Hamiltonian, projected density of states, and band interpolation to be expressed in a basis that separates impurity, substrate, surface, or orbital sectors (0910.1748). In later developments, analogous sectoring appears in symmetry-quantized formal polarization, band representations, fragile topology, reduced Wannier representations, and response decompositions, where the relevant fragment may be a unit-rank band, a frozen subspace inside an entangled manifold, a trivial Wannierizable component of a topological set of bands, or an optically connected pair of fractional Wannier-center sectors (Pang et al., 15 Jun 2026).

1. Formal basis of Wannier-sector resolution

The basic construction starts from Bloch eigenstates $\psi_{\mathbf{k}n}(\mathbf{r})$ and defines Wannier functions by a $\mathbf{k}$-dependent unitary mixing,

$$w_m(\mathbf{r}-\mathbf{R})=\frac{1}{\Omega^*}\int_{BZ}\sum_n U_{mn}(\mathbf{k})\psi_{\mathbf{k}n}(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{R}}\,d^3\mathbf{k},$$

with $U_{mn}(\mathbf{k})$ chosen to minimize the Marzari-Vanderbilt spread functional

$$\Omega=\sum_m\left[\langle r^2\rangle_m-\langle \mathbf{r}\rangle_m^2\right].$$

This yields maximally localized Wannier functions that are orthonormal and spatially compact (0910.1748).

Once such a basis is available, fragmentation becomes explicit at the level of both states and observables. In the Co-in-Cu application, the Hamiltonian in the Wannier representation is written in block form,

$$\begin{pmatrix} \mathcal{H}_{subs} & \mathcal{V}\ \mathcal{V}^\dagger & \mathcal{H}_{imp} \end{pmatrix},$$

where $\mathcal{H}_{subs}$ describes host Wannier functions, $\mathcal{H}_{imp}$ impurity Wannier functions, and $\mathcal{V}$ their hybridization. The same construction supports a Wannier-projected density of states,

$$\rho_{m\sigma}(\omega)=\sum_{n\mathbf{k}}|U_{mn}(\mathbf{k}\sigma)|^2\delta(\omega-\epsilon_{n\mathbf{k}\sigma}),$$

and occupations

$\mathbf{k}$0

These quantities resolve how a chosen Wannier orbital contributes across energy, so that Co $\mathbf{k}$1, Co $\mathbf{k}$2, Cu $\mathbf{k}$3, surface, and bulk sectors can be identified separately (0910.1748).

In this sense, fragmentation is not merely localization. It is a controlled rewriting of the spectral problem into sectors whose real-space character, band weight, and hybridization are directly computable. The Cu(111) surface example illustrates the point: the Shockley surface state is captured in a Wannier basis that distinguishes surface and bulk components, and its character can be followed across $\mathbf{k}$4 as the state hybridizes with bulk bands (0910.1748).

2. Disentanglement, frozen subspaces, and sector selection in entangled bands

For metals and surfaces, the relevant bands are typically entangled rather than isolated. The practical solution is the Souza-Marzari-Vanderbilt disentanglement scheme: one chooses an outer window of candidate states, optionally an inner or frozen window to be reproduced exactly, and trial orbitals $\mathbf{k}$5 that encode the desired orbital content. The basic overlap data are

$\mathbf{k}$6

from which an optimal subspace is selected variationally (0910.1748).

In the variational formulation for entangled bands, the Wannier subspace projector $\mathbf{k}$7 is allowed to be larger than the spectral subspace of immediate interest, but it must contain the frozen subspace projector $\mathbf{k}$8 exactly:

$\mathbf{k}$9

This constraint prevents the target bands from being split fractionally among incompatible Wannier sectors. The resulting optimization problem minimizes the spread over matrices $$w_m(\mathbf{r}-\mathbf{R})=\frac{1}{\Omega^*}\int_{BZ}\sum_n U_{mn}(\mathbf{k})\psi_{\mathbf{k}n}(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{R}}\,d^3\mathbf{k},$$0 subject to orthonormality and inclusion of the frozen bands. In this framework, the standard disentanglement algorithm is interpreted as a splitting method that alternates between optimizing the projector and optimizing the gauge, rather than performing a joint minimization (Damle et al., 2018).

This distinction matters for localization and for the sharpness of fragmentation. With an SCDM initial guess, the variational formulation robustly finds more localized Wannier functions for silicon, copper, and aluminum, while preserving the required frozen-band content. In model free-electron gases, the maximally localized Wannier functions for entangled bands decay only algebraically, and super-algebraic decay requires further gauge smoothing or higher-moment optimization (Damle et al., 2018). A plausible implication is that band-resolved fragmentation in entangled systems is controlled as much by subspace regularity as by the formal existence of a Wannier basis.

3. Symmetry, band representations, and topological obstruction

A second meaning of Wannier-sector fragmentation appears in the theory of band representations. The crystallographic splitting theorem states that a rank-$$w_m(\mathbf{r}-\mathbf{R})=\frac{1}{\Omega^*}\int_{BZ}\sum_n U_{mn}(\mathbf{k})\psi_{\mathbf{k}n}(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{R}}\,d^3\mathbf{k},$$1 band representation is equivalent to a decomposition

$$w_m(\mathbf{r}-\mathbf{R})=\frac{1}{\Omega^*}\int_{BZ}\sum_n U_{mn}(\mathbf{k})\psi_{\mathbf{k}n}(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{R}}\,d^3\mathbf{k},$$2

into $$w_m(\mathbf{r}-\mathbf{R})=\frac{1}{\Omega^*}\int_{BZ}\sum_n U_{mn}(\mathbf{k})\psi_{\mathbf{k}n}(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{R}}\,d^3\mathbf{k},$$3 unit-rank bands such that each $$w_m(\mathbf{r}-\mathbf{R})=\frac{1}{\Omega^*}\int_{BZ}\sum_n U_{mn}(\mathbf{k})\psi_{\mathbf{k}n}(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{R}}\,d^3\mathbf{k},$$4 is analytic and any symmetry acts by permuting the sector labels $$w_m(\mathbf{r}-\mathbf{R})=\frac{1}{\Omega^*}\int_{BZ}\sum_n U_{mn}(\mathbf{k})\psi_{\mathbf{k}n}(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{R}}\,d^3\mathbf{k},$$5 (Alexandradinata et al., 2019). Here fragmentation is exact and symmetry controlled: a multi-rank band representation can be resolved into unit-rank Wannier sectors whenever the required analytic and symmetry-permuted splitting exists.

This perspective sharpens the distinction between atomicity, obstruction, and fragility. Fragile topology is defined by a set of bands that is not Wannier-representable by itself but becomes Wannier-representable after additional trivial degrees of freedom are added (Po et al., 2017). In the explicit honeycomb example, a split elementary band representation produces trivial valence bands with symmetric exponentially localized Wannier functions and conduction bands with a Wannier obstruction; the obstruction disappears when the two are recombined into the full atomic four-band set (Po et al., 2017). Fragmentation therefore need not divide a system into “topological” pieces on both sides of a gap.

Twisted bilayer graphene makes the same issue concrete in a symmetry-diagnostic form. For a single valley, the nearly flat two-band sector has a Wannier obstruction associated with the coexistence of opposite mirror eigenvalues at high-symmetry points and same-chirality Dirac nodes; the paper shows these two obstructions are equivalent. The obstruction is resolved in a symmetry-faithful four-band model per valley, where the lower two bands reproduce the same obstruction but the enlarged Hilbert space admits a tight-binding construction (Zou et al., 2018). This is a fragile, not stable, obstruction.

Band connectivity further constrains whether an apparent splitting really defines invariant Wannier sectors. The transformation properties of a band representation depend on band connectivity, and EBRs arising from Wyckoff positions with multiplicity can admit multiple atomic limits. Symmetry-permitted split EBRs that do not correspond to connected irreducible band representations are potentially non-trivial on both sides of the gap (Zhang, 2021). This suggests that fragmentation must be distinguished from mere energetic separation: the decisive question is whether the split sectors remain symmetry-invariant across the Brillouin zone.

4. Transition-resolved fragmentation and optical or magnetic response

A third usage of fragmentation is transition resolved rather than state resolved. In quantized formal polarization crystals, symmetry-quantized formal-polarization branches correspond to fractional Wannier-center sectors. The band-resolved Berry-phase polarization of an isolated band is

$$w_m(\mathbf{r}-\mathbf{R})=\frac{1}{\Omega^*}\int_{BZ}\sum_n U_{mn}(\mathbf{k})\psi_{\mathbf{k}n}(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{R}}\,d^3\mathbf{k},$$6

and optical transitions between bands $$w_m(\mathbf{r}-\mathbf{R})=\frac{1}{\Omega^*}\int_{BZ}\sum_n U_{mn}(\mathbf{k})\psi_{\mathbf{k}n}(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{R}}\,d^3\mathbf{k},$$7 and $$w_m(\mathbf{r}-\mathbf{R})=\frac{1}{\Omega^*}\int_{BZ}\sum_n U_{mn}(\mathbf{k})\psi_{\mathbf{k}n}(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{R}}\,d^3\mathbf{k},$$8 are governed by the difference

$$w_m(\mathbf{r}-\mathbf{R})=\frac{1}{\Omega^*}\int_{BZ}\sum_n U_{mn}(\mathbf{k})\psi_{\mathbf{k}n}(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{R}}\,d^3\mathbf{k},$$9

The shift vector obeys

$U_{mn}(\mathbf{k})$0

Accordingly, a transition between occupied and low-lying unoccupied states whose Wannier centers lie at distinct fractional Wyckoff positions produces a large transition-resolved Wannier-center displacement, a large shift vector, and a dominant shift-vector-related intraband contribution to the static SHG susceptibility (Pang et al., 15 Jun 2026).

In this formulation, Wannier-sector fragmentation means that bands near the gap are grouped into subsets with distinct symmetry-protected fractional Wannier centers, and optically allowed transitions cross sector boundaries. The paper emphasizes that quantized formal polarization alone is not sufficient for a large nonlinear response; what matters is that optically active transitions connect distinct fractional sectors (Pang et al., 15 Jun 2026).

An analogous decomposition exists for orbital magnetic susceptibility. Using modified Wannier functions, the susceptibility is resolved into

$U_{mn}(\mathbf{k})$1

with $U_{mn}(\mathbf{k})$2–$U_{mn}(\mathbf{k})$3 intraband and $U_{mn}(\mathbf{k})$4–$U_{mn}(\mathbf{k})$5 interband, and with an independent classification into itinerant and local contributions. Inside a gap at $U_{mn}(\mathbf{k})$6, only $U_{mn}(\mathbf{k})$7, $U_{mn}(\mathbf{k})$8, and $U_{mn}(\mathbf{k})$9 survive. The paper further shows that the sharpness of this decomposition depends on the localization of the underlying Wannier functions: poor localization enhances intergroup terms and weakens the band-specific interpretation (Kariyado et al., 2021). Fragmentation is therefore operational only to the extent that a localized sector basis exists.

5. Reduced, dual, and automated Wannier representations

When full Wannierization is topologically obstructed, several generalized constructions produce partial or mixed forms of fragmentation. The reduced Wannier representation decomposes a topological manifold into a localized trivial subspace and an itinerant topological complement,

$$\Omega=\sum_m\left[\langle r^2\rangle_m-\langle \mathbf{r}\rangle_m^2\right].$$0

with a Wannier fraction

$$\Omega=\sum_m\left[\langle r^2\rangle_m-\langle \mathbf{r}\rangle_m^2\right].$$1

In supercells, this procedure yields exponentially localized Wannier functions spanning a proper subspace of the topological manifold while confining the obstruction to a smaller residual sector. In the Haldane model, for example, a $$\Omega=\sum_m\left[\langle r^2\rangle_m-\langle \mathbf{r}\rangle_m^2\right].$$2 supercell supports three localized reduced Wannier functions out of four occupied bands, leaving a one-dimensional topological complement; larger supercells increase $$\Omega=\sum_m\left[\langle r^2\rangle_m-\langle \mathbf{r}\rangle_m^2\right].$$3 toward $$\Omega=\sum_m\left[\langle r^2\rangle_m-\langle \mathbf{r}\rangle_m^2\right].$$4 without eliminating the obstruction (Cole et al., 2024).

A different generalization seeks simultaneous localization in space and energy. Dually localized Wannier functions minimize

$$\Omega=\sum_m\left[\langle r^2\rangle_m-\langle \mathbf{r}\rangle_m^2\right].$$5

where $$\Omega=\sum_m\left[\langle r^2\rangle_m-\langle \mathbf{r}\rangle_m^2\right].$$6 is the spatial variance and $$\Omega=\sum_m\left[\langle r^2\rangle_m-\langle \mathbf{r}\rangle_m^2\right].$$7 the energy variance. Because valence and conduction bands may be localized together, these functions naturally associate individual Wannier orbitals with specific energy ranges and induce fractional occupations through the non-diagonal density matrix in the DLWF basis. In silicon and ethylene, this yields bonding and antibonding frontier orbitals; in copper, it yields localized $$\Omega=\sum_m\left[\langle r^2\rangle_m-\langle \mathbf{r}\rangle_m^2\right].$$8, $$\Omega=\sum_m\left[\langle r^2\rangle_m-\langle \mathbf{r}\rangle_m^2\right].$$9, and $$\begin{pmatrix} \mathcal{H}_{subs} & \mathcal{V}\ \mathcal{V}^\dagger & \mathcal{H}_{imp} \end{pmatrix},$$0 character together with a more extended frontier orbital near the Fermi level (Mahler et al., 2022). Here the fragmentation is explicitly energy resolved.

Automation addresses yet another difficulty: topologically nontrivial bands may admit localized Wannier functions only after non-obvious symmetry breaking and orbital mixing. The optimized projection functions method uses a redundant set of projection orbitals and a semiunitary optimization to construct maximally localized Wannier functions automatically for $$\begin{pmatrix} \mathcal{H}_{subs} & \mathcal{V}\ \mathcal{V}^\dagger & \mathcal{H}_{imp} \end{pmatrix},$$1 topological insulators and for Bi$$\begin{pmatrix} \mathcal{H}_{subs} & \mathcal{V}\ \mathcal{V}^\dagger & \mathcal{H}_{imp} \end{pmatrix},$$2Se$$\begin{pmatrix} \mathcal{H}_{subs} & \mathcal{V}\ \mathcal{V}^\dagger & \mathcal{H}_{imp} \end{pmatrix},$$3. In all cases, the resulting Wannier functions contain large imaginary components and are more extended than those in the topologically trivial phase (Mustafa et al., 2016). This does not remove the topological content; it exposes the real-space form that the residual obstruction forces onto the Wannier sectors.

6. Disorder, dynamical systems, and limitations of single-band sectoring

Wannier-sector fragmentation is not restricted to clean electronic crystals. In the disordered $$\begin{pmatrix} \mathcal{H}_{subs} & \mathcal{V}\ \mathcal{V}^\dagger & \mathcal{H}_{imp} \end{pmatrix},$$4-flux ladder, the relevant sectors are the eigenstates of the Wannier Hamiltonian $$\begin{pmatrix} \mathcal{H}_{subs} & \mathcal{V}\ \mathcal{V}^\dagger & \mathcal{H}_{imp} \end{pmatrix},$$5, which define upper and lower Wannier bands with polarizations quantized to $$\begin{pmatrix} \mathcal{H}_{subs} & \mathcal{V}\ \mathcal{V}^\dagger & \mathcal{H}_{imp} \end{pmatrix},$$6 or $$\begin{pmatrix} \mathcal{H}_{subs} & \mathcal{V}\ \mathcal{V}^\dagger & \mathcal{H}_{imp} \end{pmatrix},$$7. Disorder can drive a Wannier-band transition when the Wannier gap closes even while the energy gap remains open. The real-space renormalization-group analysis shows that local rearrangements of plaquettes do not change the topological invariant; the transition requires a nonlocal event in which a plaquette stretches to half the system size, and the Wannier localization length diverges at the transition (Claes et al., 2020). This is a direct example of sector-resolved topology changing without an ordinary bulk energy-gap closing.

The same paper establishes a correspondence between Wannier-sector topology and the energy-band topology of a physically cut system: cutting the vertical bonds maps the ladder to two SSH chains, and the topology of each Wannier sector matches the topology of the corresponding chain, even in the presence of disorder (Claes et al., 2020). This clarifies that fragmented Wannier sectors can encode physically observable boundary polarization, not merely auxiliary computational structure.

Dynamical bosonic systems reveal a complementary limitation. For a Bose-Einstein condensate in an optical cavity, a single-band Wannier expansion,

$$\begin{pmatrix} \mathcal{H}_{subs} & \mathcal{V}\ \mathcal{V}^\dagger & \mathcal{H}_{imp} \end{pmatrix},$$8

captures only the lowest-band site-resolved fragmentation, whereas a position-basis expansion,

$$\begin{pmatrix} \mathcal{H}_{subs} & \mathcal{V}\ \mathcal{V}^\dagger & \mathcal{H}_{imp} \end{pmatrix},$$9

resolves higher momentum sectors and higher Bloch bands. Both bases qualitatively reproduce fragmentation dynamics in the density-wave phase, but matter-wave superradiance is absent in the single-band Wannier basis, and the two descriptions predict different limit cycles because the single-band expansion cannot represent the necessary higher-band processes (Sevilla et al., 11 Apr 2025). The paper therefore identifies a sharp boundary for Wannier-sector fragmentation: in multiband driven systems, a single-band sector decomposition is incomplete by construction.

Taken together, these developments define band-resolved Wannier-sector fragmentation as a family of closely related procedures for isolating physically meaningful subspaces inside composite band structures. In weakly correlated metallic and surface systems it provides orbital and hybridization resolution; in topological band theory it distinguishes atomic, fragile, and obstructed decompositions; in nonlinear optics it converts symmetry-quantized formal polarization into transition-resolved Wannier-center displacement; and in disorder or dynamics it identifies sector transitions that are invisible in ordinary energy-band language. The common requirement is not simply the existence of Wannier functions, but the existence of a sector structure whose localization, symmetry, and connectivity are sufficiently controlled to support a band-resolved interpretation.