Hybrid Wannier Charge Centers

• Hybrid Wannier Charge Centers are defined as the expectation value of the position operator in a basis where states are localized along one direction and Bloch-like in others, revealing electronic topology.
• Their evolution over transverse momenta exposes key invariants such as Chern numbers, Z₂ indices, and mirror Chern numbers, linking bulk properties to observable surface and edge states.
• They are computed using Wilson loop and Berry phase methods in ab initio and tight-binding frameworks, providing robust diagnoses for a variety of topological quantum materials.

A hybrid Wannier charge center (WCC) is the expectation value of the position operator in a hybrid Wannier basis, in which Bloch states are localized in one spatial direction while remaining Bloch-like in the remaining directions. The evolution of these centers as functions of crystal momentum provides a real-space fingerprint of bulk electronic topology, capturing invariants—such as the Chern number, Z₂ indices, mirror Chern numbers, and higher multipole moments—in a wide variety of topological and crystalline phases. The WCC formalism not only enables the identification of bulk topological phase, but also provides practical, unambiguous connection between bulk invariants and physical observables such as surface or edge states, corner or hinge modes, charge pumping, and responses like the magnetoelectric (axion) effect. The computation and analysis of WCCs form a central tool in both ab initio and tight-binding approaches to topology in quantum materials.

1. Mathematical Definition and Construction

A hybrid Wannier function is obtained by performing a Fourier transform over one chosen reciprocal direction (e.g., z), yielding a state localized along that real-space direction (e.g., z), while maintaining its Bloch character in the remaining directions (e.g., x, y). For an occupied band $n$, the hybrid Wannier function for home unit cell ($l_z = 0$) is

$$W_{n0}(k_x, k_y; \vec{r}) = \frac{1}{2\pi}\int dk_z\, e^{ik_z(z-\hat{z})} u_{n, (k_x, k_y, k_z)}(\vec{r})$$

where $u_{n,\vec{k}}$ is the cell-periodic Bloch function. The hybrid Wannier charge center is then

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

Mathematically, the WCC arises as the phase (the Berry phase) of the product of overlaps along the Wannierization direction. For a one-dimensional chain (or for a hybrid corresponding to, say, $k_x$ fixed), the Berry phase

$$\phi_n(k_\perp) = -\mathrm{Im}\,\ln \prod_{k} \langle u_{n,k} | u_{n,k+\delta k} \rangle$$

yields

$$\bar{z}_n(k_\perp) = \frac{c}{2\pi} \phi_n(k_\perp)$$

In the multiband scenario, the non-Abelian Berry connection and Wilson loop formulation generalize this to obtain eigenvalues $\exp(-i\phi_n)$, which yield the centers for each band sheet (Taherinejad et al., 2013).

2. Topological Invariants from WCC Evolution

Plotting WCCs as functions of the transverse momenta (e.g., $k_x$ and $k_y$) produces continuous "sheets" over the projected 2D Brillouin zone. The global connectivity—in particular, winding and partner switching—of these sheets encodes the bulk topology:

• Chern Insulators: The WCC wind by an integer multiple of the lattice constant as $k_\perp$ traverses the Brillouin zone, corresponding to the Chern number (quantized charge pumping) (Taherinejad et al., 2013, Gresch et al., 2016).
• Z₂ topological insulators: WCCs switch partners an odd number of times along half the BZ boundary, yielding a nontrivial Z₂ index. The even/odd parity of such crossing events directly reflects $\mathbb{Z}_2$ invariants (Taherinejad et al., 2013).
• Mirror/topological crystalline insulators: WCCs can be decomposed into mirror eigenvalue sectors, and the difference in their windings on symmetry-invariant planes gives mirror Chern numbers and associated protected surface modes (Rauch et al., 2021).
• Higher-order phases/multipole insulators: The static configuration of (hybrid) Wannier centers—pinned by symmetry to Wyckoff positions—determines quantized bulk dipole, quadrupole, or octupole moments, linking directly to edge, corner, or hinge charges (Watanabe et al., 2020, Yang et al., 2022).

Table: Topological Phases and Corresponding WCC Signatures

Phase Type	WCC Evolution	Invariant
Chern insulator	Integer winding	Chern number
Z₂ topological insulator	Partner switching	Z₂ index
Topological crystalline ins.	Mirror-sector windings	Mirror Chern number
Higher-order insulator	Fixed/pinned WCC positions	Multipole moment (e.g., $q_{xy}$)

The explicit analysis of WCC sheets thus replaces explicit computation of surface or edge states and allows for bulk-only diagnosis of topology, circumventing dependence on specific terminations or gauge choices (Taherinejad et al., 2013).

3. Computation and Algorithms

The practical construction of WCCs is based directly on the electronic structure:

• Tight-binding and First-principles: The overlap matrices between cell-periodic parts at adjacent $k$-points are used to create a Wilson loop; its eigenvalues give the desired Berry phases. Singular value decomposition and parallel-transport methods are employed to unambiguously define the centers (Taherinejad et al., 2013, Gresch et al., 2016).
• Software implementations: Methods based on the evolution of WCCs (sometimes called hybrid Wannier charge centers or HWCCs) are implemented in codes such as Z2Pack (Gresch et al., 2016), WIEN2k (Gomez-Bastidas et al., 2023), and modern versions of Wannier90, WannierTools, or via software specific to all-electron/full-potential DFT methods (Tillack et al., 2019).
• Universality: The WCC approach is compatible with $k\cdot p$ models, tight-binding Hamiltonians, and ab initio calculations, and is robust to the presence or absence of inversion or mirror symmetry (Gresch et al., 2016, Rauch et al., 2021).

Notably, the methods do not require constructing fully maximally-localized Wannier functions. They operate directly in the Bloch basis by integrating gauge-invariant Berry connections.

4. Physical Consequences and Experimental Signatures

The evolution of hybrid Wannier charge centers connects directly to several measurable phenomena:

• Bulk-boundary correspondence: The winding or partner switching pattern of WCCs predicts the required existence and character of surface or edge states; gapless surface Dirac cones are mirrored by the nontrivial "flow" of bulk WCC sheets (Taherinejad et al., 2013).
• Chiral magnetic effect: In 3D Dirac/Weyl semimetals, the spiral structure of the WCC sheets is associated with the anomalous transport attributed to the chiral magnetic effect. The winding number of the WCC spiral acts as a topological monopole charge that appears in expressions for the chiral current (Osada, 10 Aug 2025).
• Higher multipole and corner charges: The quantized arrangement of WCCs in higher-order topological insulators is predictive of quantized corner or hinge charges (Watanabe et al., 2020, Yang et al., 2022).
• Spectroscopic measurement: The real-space projection of the charge density via STM/STS, integrating over a given band, allows experimental mapping of Wannier centers as peaks in the charge distribution. This has been demonstrated in artificial 1D lattices and serves as a direct experimental identification of topological and trivial phases (Ligthart et al., 19 Jul 2024).

5. Applications and Generalizations

Hybrid Wannier charge center methods extend to a range of systems and contexts:

• Crystalline/topological insulators: The WCC formalism naturally incorporates mirror and other crystalline symmetries, facilitating calculations of mirror Chern numbers and magnetoelectric axion response (Varnava et al., 2019, Rauch et al., 2021).
• Floquet/topological pumping: In periodically driven (Floquet) systems, the hybrid Wannier centers generalize to functions of both momentum and time, providing a framework for diagnosing anomalous and micromotion-induced topological phases (Nakagawa et al., 2019).
• Multipole/topological insulators: Nested Wilson loop analysis of WCCs quantifies, and directly connects, bulk quadrupole moments to corner charges and zero-energy corner states (Yang et al., 2022).
• Correlated materials: The analysis of Wannier centers built from DFT, with or without hybridization and disentanglement, is leveraged in strongly correlated systems and in the context of charge transfer and superexchange interactions (Scaramucci et al., 2014, Mao et al., 2023).
• Charge partitioning and bonding: Joint Wannier center (JWC) methods, derived from the geometric phase matrices at all $k$-points, map the distribution of electrons—both inter- and intra-atomic sharing—across complex and multicentered bonds (Saha et al., 24 Jul 2024).

6. Advantages and Limitations

The hybrid Wannier charge center framework provides several advantages:

• Model- and symmetry-independence: The calculation requires only bulk electronic structure; no explicit or assumed boundary conditions are needed (Taherinejad et al., 2013, Gresch et al., 2016).
• Unambiguous diagnosis of topology: The approach reveals the full suite of topological invariants in Chern, $\mathbb{Z}_2$, and crystalline phases (including "axion insulators" via analysis of symmetry-induced flows and Dirac touchings) (Varnava et al., 2019, Rauch et al., 2021).
• Connection to physical phenomena: The WCC evolution provides a direct visual and computational correspondence to physical observables—current pumping, surface/edge/corner states—and, in recently developed experimental protocols, can be measured via STS in artificial lattices (Ligthart et al., 19 Jul 2024).
• Extension to interactions and complexities: The method applies as long as the ground state is gapped and adiabatically connected to an atomic insulator, and can capture higher-multipole topologies even in the presence of many-body interactions (Watanabe et al., 2020).

However, certain limitations persist: when multiple directions require localization, a unique analytic gauge may not be globally attainable, and for systems without well-separated bands, the interpretation of the WCC flow can become nontrivial.

7. Future Directions and Open Problems

Several emerging research directions are rooted in WCC formalism:

• Floquet and time-dependent systems: Generalizations to time-dependent and frequency-domain Wannier centers for classification of periodically driven/dynamically modulated topological phases (Nakagawa et al., 2019).
• Finite- and higher-dimensional systems: Systematic connection between WCC flow and quantized responses, including the half-quantized surface anomalous Hall conductivity in axion insulators, and bulk-corner correspondence in multipole insulators (Varnava et al., 2019, Watanabe et al., 2020).
• Joint localization schemes: Methods based on geometric-phase-derived joint Wannier centers allow unbiased charge partitioning in both isolated and periodic systems, matching chemical bonding intuition and providing direct population analysis (Saha et al., 24 Jul 2024).
• Direct experimental measurement: STM/STS-based Wannier center spectroscopy provides a direct spatial probe of bulk topology in engineered lattices, bridging the gap between abstract invariants and observable properties (Ligthart et al., 19 Jul 2024).

In summary, hybrid Wannier charge centers form a unifying real-space framework to identify, classify, and interrogate the topology of quantum materials, encompassing both traditional (Chern, $\mathbb{Z}_2$) and higher-order (multipole, axion) invariants, connecting theory, computation, and experiment across a broad spectrum of condensed matter physics.