Hybrid Wannier Charge Centers

• Hybrid Wannier Charge Centers are real-space centers of hybrid Wannier functions that encapsulate Berry phases and topological invariants.
• They are computed via Wilson loops and overlap matrices along a crystallographic direction, enabling momentum-resolved polarization analyses.
• Their evolution provides insights into electronic structure, symmetry-protected topology, and extensions to axion and higher-order topological phases.

Hybrid Wannier Charge Centers (WCCs) are the real-space centers of hybrid Wannier functions, namely states that are Wannier-localized along one crystallographic direction while remaining Bloch-like in the others. In contemporary band theory, they are equivalently the eigenphases of a non-Abelian Wilson loop, or Berry phases along a closed reciprocal-space cycle, expressed as positions modulo a lattice vector. Their evolution over a projected Brillouin zone provides a bulk representation of electric polarization, Chern and $\mathbb Z_2$ topology, crystalline topological indices, and, in several extensions, axion response, higher-order topology, chemical bonding, and optical geometry (Taherinejad et al., 2013, Gresch et al., 2016, Varnava et al., 2019).

1. Formal definition and geometric meaning

Choosing $z$ as the Wannierized direction, hybrid Wannier functions may be written as

$$|W_{n l_z}(k_x,k_y)\rangle = \frac{1}{2\pi}\int dk_z\, e^{i\mathbf{k}\cdot(\mathbf{r}-l_z c\,\hat z)} |u_{n\mathbf{k}}\rangle ,$$

or, equivalently in the notation used for 3D insulators,

$$|h_{ln\mathbf{k}}\rangle=\frac{1}{2\pi} \int^{\pi/c}_{-\pi/c} dk_z \, e^{-ik_z l c} \ket{\psi_{n\mathbf{k}k_z}} .$$

The corresponding WCC is the position expectation value

$$\bar z_n(k_x,k_y)=\langle W_{n0}|\hat z|W_{n0}\rangle ,$$

or, equivalently,

$z_{ln}(\mathbf{k})=\bra{h_{ln\mathbf{k}}z\ket{h_{ln\mathbf{k}} , \qquad z_{ln}(\mathbf{k})=z_{0n}(\mathbf{k})+lc .$

In this representation, the WCCs are the eigenvalues of the projected position operator $P z P$, and their dependence on the transverse momentum $(k_x,k_y)$ defines “Wannier bands” or “WCC sheets” over the projected 2D Brillouin zone (Taherinejad et al., 2013, Varnava et al., 2019, Rauch et al., 2021).

For an isolated band, the WCC is the Berry phase along the Wannierized direction written as a real-space coordinate: $$\bar z(k_\perp)=\frac{c}{2\pi}\phi(k_\perp), \qquad \phi=-\mathrm{Im}\,\ln \prod \langle u_k|u_{k+b}\rangle .$$ In the multiband case, the same object appears as a Wilson-loop eigenphase. If the Wilson-loop eigenvalues are $e^{i\varphi_n(\mathbf{k})}$, then

$z$0

This equivalence underlies the standard identification of hybrid WCCs with Wilson-loop spectra (Taherinejad et al., 2013, Varnava et al., 2019).

The physical interpretation is twofold. First, the sum of WCCs gives a polarization-like quantity, so WCC flow is a momentum-resolved form of Berry-phase polarization. Second, because WCCs are defined modulo a lattice vector, only their global evolution, winding, and connectivity are topologically meaningful. This is why WCC plots are typically read as spectral-flow diagrams rather than as absolute positions (Gresch et al., 2016, Gomez-Bastidas et al., 2023).

2. Wilson loops, overlap matrices, and numerical construction

In practical calculations, hybrid WCCs are extracted from overlap matrices between neighboring $z$1-points on a closed loop. For a chosen band subspace,

$z$2

and the cumulative loop matrix is

$z$3

Its eigenvalues $z$4 encode the Berry phases, and the hybrid WCCs are

$z$5

This is the Wilson-loop/HWCC construction implemented directly from first-principles Bloch states in the all-electron WIEN2k workflow, where no maximally localized Wannier-function construction is needed (Gomez-Bastidas et al., 2023).

A closely related formulation used in Z2Pack proceeds by parallel transport. One computes overlap matrices

$z$6

performs a singular value decomposition $z$7, forms the unitary transport step $z$8, and multiplies these along the loop. If the resulting non-Abelian phase matrix has eigenvalues $z$9, the normalized WCCs are

$$|W_{n l_z}(k_x,k_y)\rangle = \frac{1}{2\pi}\int dk_z\, e^{i\mathbf{k}\cdot(\mathbf{r}-l_z c\,\hat z)} |u_{n\mathbf{k}}\rangle ,$$0

up to orientation convention (Gresch et al., 2016).

These constructions operate on the occupied subspace as a whole. The gauge freedom is handled implicitly by working with overlap matrices and Wilson loops rather than by differentiating phases of individual bands. This makes the method numerically stable and naturally suited to composite occupied manifolds, including all-electron, tight-binding, and $$|W_{n l_z}(k_x,k_y)\rangle = \frac{1}{2\pi}\int dk_z\, e^{i\mathbf{k}\cdot(\mathbf{r}-l_z c\,\hat z)} |u_{n\mathbf{k}}\rangle ,$$1 calculations (Gresch et al., 2016, Gomez-Bastidas et al., 2023).

Several practical caveats are standard. The phases are defined modulo $$|W_{n l_z}(k_x,k_y)\rangle = \frac{1}{2\pi}\int dk_z\, e^{i\mathbf{k}\cdot(\mathbf{r}-l_z c\,\hat z)} |u_{n\mathbf{k}}\rangle ,$$2, so WCCs are defined modulo a lattice vector or modulo $$|W_{n l_z}(k_x,k_y)\rangle = \frac{1}{2\pi}\int dk_z\, e^{i\mathbf{k}\cdot(\mathbf{r}-l_z c\,\hat z)} |u_{n\mathbf{k}}\rangle ,$$3 in fractional coordinates. The physically meaningful quantity is therefore the continuous evolution of the WCC spectrum as the transverse momentum varies. The WIEN2k implementation also emphasizes that there is no exact one-to-one mapping between original Bloch-band indices and WCC indices, because the WCCs arise from diagonalizing the Wilson-loop matrix for the entire occupied subspace (Gomez-Bastidas et al., 2023).

3. Topological diagnostics from WCC flow

The principal use of hybrid WCCs is topological diagnosis through spectral flow. For Chern insulators, the Chern number is the net change of the summed WCCs over a pumping cycle: $$|W_{n l_z}(k_x,k_y)\rangle = \frac{1}{2\pi}\int dk_z\, e^{i\mathbf{k}\cdot(\mathbf{r}-l_z c\,\hat z)} |u_{n\mathbf{k}}\rangle ,$$4 or equivalently

$$|W_{n l_z}(k_x,k_y)\rangle = \frac{1}{2\pi}\int dk_z\, e^{i\mathbf{k}\cdot(\mathbf{r}-l_z c\,\hat z)} |u_{n\mathbf{k}}\rangle ,$$5

This is the statement that the Chern number is a quantized charge pump encoded in WCC winding (Gresch et al., 2016).

For time-reversal-invariant insulators, the relevant invariant is the parity of WCC crossings over half the Brillouin zone. In practice one follows the largest gap in the WCC spectrum and counts whether a Wannier band is crossed an odd or even number of times. The WIEN2k implementation summarizes the criterion as follows: if “one Wannier band is crossed when following the largest gap in the HWCC spectrum,” the phase is nontrivial. This is the Wilson-loop version of partner switching or adiabatic Thouless charge pumping (Gomez-Bastidas et al., 2023, Gresch et al., 2016).

In three dimensions, WCC sheets play the same diagnostic role as surface-state dispersions, but are obtained from bulk calculations alone. The “Wannier Center Sheets in Topological Insulators” framework shows that trivial, weak, strong, and crystalline topological insulators can be distinguished by the connectivity of WCC sheets over projected 2D Brillouin zones. In first-principles examples, Sb$$|W_{n l_z}(k_x,k_y)\rangle = \frac{1}{2\pi}\int dk_z\, e^{i\mathbf{k}\cdot(\mathbf{r}-l_z c\,\hat z)} |u_{n\mathbf{k}}\rangle ,$$6Se$$|W_{n l_z}(k_x,k_y)\rangle = \frac{1}{2\pi}\int dk_z\, e^{i\mathbf{k}\cdot(\mathbf{r}-l_z c\,\hat z)} |u_{n\mathbf{k}}\rangle ,$$7 exhibits topologically trivial WCC sheets, KHgSb exhibits weak-topological behavior, and Bi$$|W_{n l_z}(k_x,k_y)\rangle = \frac{1}{2\pi}\int dk_z\, e^{i\mathbf{k}\cdot(\mathbf{r}-l_z c\,\hat z)} |u_{n\mathbf{k}}\rangle ,$$8Se$$|W_{n l_z}(k_x,k_y)\rangle = \frac{1}{2\pi}\int dk_z\, e^{i\mathbf{k}\cdot(\mathbf{r}-l_z c\,\hat z)} |u_{n\mathbf{k}}\rangle ,$$9 exhibits strong-topological behavior (Taherinejad et al., 2013).

The same logic extends to semimetals when the chosen manifold remains gapped. Z2Pack uses WCC winding on a small sphere to determine the chirality of a Weyl node, and can detect Weyl or Dirac structure from changes of Chern or $$|h_{ln\mathbf{k}}\rangle=\frac{1}{2\pi} \int^{\pi/c}_{-\pi/c} dk_z \, e^{-ik_z l c} \ket{\psi_{n\mathbf{k}k_z}} .$$0 invariants between planes or curved surfaces in the Brillouin zone (Gresch et al., 2016). In layered organic conductors, WCC sheets localized along $$|h_{ln\mathbf{k}}\rangle=\frac{1}{2\pi} \int^{\pi/c}_{-\pi/c} dk_z \, e^{-ik_z l c} \ket{\psi_{n\mathbf{k}k_z}} .$$1 and plotted over $$|h_{ln\mathbf{k}}\rangle=\frac{1}{2\pi} \int^{\pi/c}_{-\pi/c} dk_z \, e^{-ik_z l c} \ket{\psi_{n\mathbf{k}k_z}} .$$2 acquire especially clear geometric forms: flat but discontinuously cut sheets for nodal-line semimetals, oppositely winding double spirals around Dirac points, same-sense spirals around Weyl points, and partner switching with $$|h_{ln\mathbf{k}}\rangle=\frac{1}{2\pi} \int^{\pi/c}_{-\pi/c} dk_z \, e^{-ik_z l c} \ket{\psi_{n\mathbf{k}k_z}} .$$3 for the weak topological-insulator phase (Osada, 10 Aug 2025).

4. Crystalline, axion, and higher-order extensions

Mirror symmetry admits a symmetry-resolved WCC analysis in which the occupied space is split into mirror eigenspaces on a mirror-invariant plane. In the hybrid-Wannier representation, the mirror Chern number can be determined from the winding numbers of touching points between Wannier bands on mirror-pinned planes and from the Chern numbers of flat Wannier bands pinned to those planes. For type-1 mirrors, the mirror Chern numbers on the $$|h_{ln\mathbf{k}}\rangle=\frac{1}{2\pi} \int^{\pi/c}_{-\pi/c} dk_z \, e^{-ik_z l c} \ket{\psi_{n\mathbf{k}k_z}} .$$4 and $$|h_{ln\mathbf{k}}\rangle=\frac{1}{2\pi} \int^{\pi/c}_{-\pi/c} dk_z \, e^{-ik_z l c} \ket{\psi_{n\mathbf{k}k_z}} .$$5 planes are

$$|h_{ln\mathbf{k}}\rangle=\frac{1}{2\pi} \int^{\pi/c}_{-\pi/c} dk_z \, e^{-ik_z l c} \ket{\psi_{n\mathbf{k}k_z}} .$$6

$$|h_{ln\mathbf{k}}\rangle=\frac{1}{2\pi} \int^{\pi/c}_{-\pi/c} dk_z \, e^{-ik_z l c} \ket{\psi_{n\mathbf{k}k_z}} .$$7

while for a type-2 mirror

$$|h_{ln\mathbf{k}}\rangle=\frac{1}{2\pi} \int^{\pi/c}_{-\pi/c} dk_z \, e^{-ik_z l c} \ket{\psi_{n\mathbf{k}k_z}} .$$8

This formalism was illustrated for SnTe in both monolayer and bulk forms (Rauch et al., 2021).

Axion or Chern-Simons topology can likewise be reformulated in WCC language. In the hybrid-Wannier representation, the axion coupling $$|h_{ln\mathbf{k}}\rangle=\frac{1}{2\pi} \int^{\pi/c}_{-\pi/c} dk_z \, e^{-ik_z l c} \ket{\psi_{n\mathbf{k}k_z}} .$$9 admits a decomposition

$$\bar z_n(k_x,k_y)=\langle W_{n0}|\hat z|W_{n0}\rangle ,$$0

with

$$\bar z_n(k_x,k_y)=\langle W_{n0}|\hat z|W_{n0}\rangle ,$$1

For axion-quantizing symmetries, the WCC sheet geometry determines when $$\bar z_n(k_x,k_y)=\langle W_{n0}|\hat z|W_{n0}\rangle ,$$2 or $$\bar z_n(k_x,k_y)=\langle W_{n0}|\hat z|W_{n0}\rangle ,$$3, and connected or disconnected Wannier spectra can encode the same axion-$$\bar z_n(k_x,k_y)=\langle W_{n0}|\hat z|W_{n0}\rangle ,$$4 index depending on whether the protecting symmetry is $$\bar z_n(k_x,k_y)=\langle W_{n0}|\hat z|W_{n0}\rangle ,$$5-preserving or $$\bar z_n(k_x,k_y)=\langle W_{n0}|\hat z|W_{n0}\rangle ,$$6-reversing (Varnava et al., 2019). In the dynamical setting of Chern-Simons axion pumping, a nonzero second Chern number requires sheet-touching events at which $$\bar z_n(k_x,k_y)=\langle W_{n0}|\hat z|W_{n0}\rangle ,$$7 quanta of Berry curvature are transferred from one WCC sheet to another, yielding $$\bar z_n(k_x,k_y)=\langle W_{n0}|\hat z|W_{n0}\rangle ,$$8 over the cycle (Taherinejad et al., 2014).

Higher-order topology introduces an additional layer. In the generalized separable BBH model, the hybrid WCCs are exactly the Wilson-loop eigenphases $$\bar z_n(k_x,k_y)=\langle W_{n0}|\hat z|W_{n0}\rangle ,$$9, and the nested-Wilson-loop structure yields exact Wannier-sector polarizations

$z_{ln}(\mathbf{k})=\bra{h_{ln\mathbf{k}}z\ket{h_{ln\mathbf{k}} , \qquad z_{ln}(\mathbf{k})=z_{0n}(\mathbf{k})+lc .$0

and quadrupole moment

$z_{ln}(\mathbf{k})=\bra{h_{ln\mathbf{k}}z\ket{h_{ln\mathbf{k}} , \qquad z_{ln}(\mathbf{k})=z_{0n}(\mathbf{k})+lc .$1

This establishes a $z_{ln}(\mathbf{k})=\bra{h_{ln\mathbf{k}}z\ket{h_{ln\mathbf{k}} , \qquad z_{ln}(\mathbf{k})=z_{0n}(\mathbf{k})+lc .$2 Wannier topology controlled by the winding numbers of the constituent one-dimensional chiral chains (Yang et al., 2022).

At the same time, ordinary hybrid WCC flow is not universally sufficient for higher-order phases. In the strained Kane–Mele model, WCC evolution clearly diagnoses the quantum spin Hall phase for $z_{ln}(\mathbf{k})=\bra{h_{ln\mathbf{k}}z\ket{h_{ln\mathbf{k}} , \qquad z_{ln}(\mathbf{k})=z_{0n}(\mathbf{k})+lc .$3, but becomes topologically trivial in the second-order topological-insulator phase for $z_{ln}(\mathbf{k})=\bra{h_{ln\mathbf{k}}z\ket{h_{ln\mathbf{k}} , \qquad z_{ln}(\mathbf{k})=z_{0n}(\mathbf{k})+lc .$4. The paper explicitly concludes that ordinary WCC evolution is “incapable of capturing any essence of the second order topological phase,” and instead uses spin-resolved bulk polarization to identify the obstructed atomic-insulator character of the SOTI phase (Lahiri et al., 2023).

5. Generalizations beyond standard WCC flow

A notable three-dimensional extension is the “joint Wannier center” construction for charge partitioning in matter. Starting from the same non-Abelian Berry-phase matrices that define hybrid WCCs along one direction,

$z_{ln}(\mathbf{k})=\bra{h_{ln\mathbf{k}}z\ket{h_{ln\mathbf{k}} , \qquad z_{ln}(\mathbf{k})=z_{0n}(\mathbf{k})+lc .$5

the method considers the three matrices $z_{ln}(\mathbf{k})=\bra{h_{ln\mathbf{k}}z\ket{h_{ln\mathbf{k}} , \qquad z_{ln}(\mathbf{k})=z_{0n}(\mathbf{k})+lc .$6, $z_{ln}(\mathbf{k})=\bra{h_{ln\mathbf{k}}z\ket{h_{ln\mathbf{k}} , \qquad z_{ln}(\mathbf{k})=z_{0n}(\mathbf{k})+lc .$7, and $z_{ln}(\mathbf{k})=\bra{h_{ln\mathbf{k}}z\ket{h_{ln\mathbf{k}} , \qquad z_{ln}(\mathbf{k})=z_{0n}(\mathbf{k})+lc .$8 associated with three linearly independent reciprocal directions and performs maximal joint diagonalization. The resulting expectation values define real-space points $z_{ln}(\mathbf{k})=\bra{h_{ln\mathbf{k}}z\ket{h_{ln\mathbf{k}} , \qquad z_{ln}(\mathbf{k})=z_{0n}(\mathbf{k})+lc .$9, called joint Wannier centers. The paper explicitly states that these are not standard hybrid WCCs, but a three-directional extension built from hybrid-Wannier data and used for chemical-bonding analysis (Saha et al., 2024).

Closely related Berry-center ideas also enter nonlinear optics. In quantized formal-polarization crystals, the relevant objects are band-resolved Berry-phase polarizations,

$P z P$0

interpreted as Wannier centers of the initial and final states. The Brillouin-zone average of the optical shift vector obeys

$P z P$1

This identifies the average shift vector with a difference of Berry-phase centers modulo a polarization quantum and winding term. In this setting the paper does not explicitly use the term “hybrid WCC,” but it uses the closely related language of Wannier centers, fractional Wannier-center sectors, and Berry-phase polarization differences between optically connected bands (Pang et al., 15 Jun 2026).

A one-dimensional experimental counterpart is Wannier-center spectroscopy in artificial lattices on Cs/InAs(111)A. There the band-resolved local density of states is integrated over energy,

$P z P$2

to reconstruct the real-space Wannier-center location. In this purely one-dimensional setting there is no transverse momentum and hence no nontrivial hybrid-WCC flow, but the method directly measures the 1D analogue of the WCC position and uses it to distinguish trivial from boundary-obstructed topological phases (Ligthart et al., 2024).

6. Implementations, representative workflows, and limitations

The most widely used general-purpose software framework in this area is Z2Pack, which implements WCC tracking for $P z P$3 models, tight-binding models, and ab initio calculations. Its workflow is based on choosing an appropriate closed manifold in the Brillouin zone, computing Wilson loops along a family of closed paths, and extracting invariants from WCC winding, parity of crossings, or symmetry-resolved subspace flow (Gresch et al., 2016). A complementary all-electron route is provided by the WIEN2k implementation built around wcc.py in BerryPI, which constructs case.klist, computes overlaps via Wien2wannier, stores them in case.mmn, and consolidates the resulting WCC spectrum into wcc.csv (Gomez-Bastidas et al., 2023).

The method is robust, but its scope is not unlimited. WCCs are naturally defined modulo a lattice vector, so branch choices matter and only continuous evolution is physically meaningful (Gomez-Bastidas et al., 2023). Dense meshes are often required when curves approach closely or when largest-gap tracking becomes ambiguous; the WIEN2k implementation recommends refining both the loop mesh and the evolution-direction mesh if the flow appears jagged or unclear (Gomez-Bastidas et al., 2023). Z2Pack likewise emphasizes convergence control, especially near small direct gaps or near topological phase transitions (Gresch et al., 2016).

Several conceptual limitations are also standard. Individual WCC branches are properties of the chosen occupied subspace and should not be interpreted as trajectories of individual Bloch bands, because the Wilson-loop eigenvectors generally mix several bands (Gomez-Bastidas et al., 2023). Standard hybrid WCCs are exact only for localization along one direction; three-directional constructions such as joint Wannier centers sacrifice exact one-directional localization in exchange for joint localization criteria (Saha et al., 2024). Ordinary WCC flow can cleanly diagnose first-order $P z P$4 topology, but may fail for higher-order phases, as in the strained Kane–Mele example (Lahiri et al., 2023).

Taken together, these developments place hybrid WCCs at the center of a broad geometric framework. In their canonical form they are Wilson-loop eigenphases interpreted as real-space charge centers; in application they act as bulk diagnostics of topology, symmetry, and adiabatic transport; and in more recent extensions they serve as building blocks for chemical partitioning, experimental charge-center spectroscopy, and Berry-geometric descriptions of optical response (Taherinejad et al., 2013, Gresch et al., 2016, Saha et al., 2024).