Wannier Tight-Binding Models

• Wannier tight-binding models are a framework that constructs effective Hamiltonians using maximally localized Wannier functions to bridge first-principles calculations and real-space descriptions.
• They accurately quantify onsite energies and extended hopping terms, ensuring that symmetry, topology, and orbital characteristics are maintained in the localized basis.
• These models are crucial in studying complex systems such as graphene, van der Waals heterostructures, and photonic crystals, enabling precise analysis of band structures and many-body effects.

Wannier tight-binding models are a foundational framework for bridging ab initio electronic structure and effective, localized descriptions of crystalline solids, artificial lattices, and engineered quantum systems. By constructing tight-binding Hamiltonians in a basis of maximally localized Wannier functions, researchers obtain models that are not only computationally efficient but also physically transparent, capturing real-space localization, symmetry, and the interplay of topology and interactions. Wannier-based parameterizations have become the standard in systematically extracting effective models from first-principles calculations, enabling quantitative studies of band structures, transport, topologically nontrivial phases, strongly correlated phenomena, and engineered properties in diverse contexts such as graphene multilayers, van der Waals heterostructures, ultracold atomic lattices, and photonic crystals.

1. Mathematical Foundations and Methodology

Construction of Wannier tight-binding models begins with the generation of Bloch states $\psi_{n\mathbf{k}}(\mathbf{r})$ from ab initio (e.g., density functional theory) or model Hamiltonians. Maximally localized Wannier functions (MLWFs), as introduced by Marzari and Vanderbilt, are defined through a unitary mixing of Bloch states within a selected energy manifold, mapped to real-space lattice centers $R_j$ and additional internal degrees of freedom (sublattice, orbital, layer, spin, etc.):

$$w_{j,\nu}(\mathbf{r}) = \frac{1}{\sqrt{V_\mathrm{BZ}}} \int_{BZ} d\mathbf{k}\,e^{-i\mathbf{k}\cdot\mathbf{R}_j} \sum_{m=1}^N U_{\nu m}(\mathbf{k})\,\psi_{m\mathbf{k}}(\mathbf{r}),$$

where $U_{\nu m}(\mathbf{k})$ is chosen to minimize the quadratic spread $$\Omega = \sum_\nu [\langle r^2 \rangle_\nu - \langle \mathbf{r} \rangle_\nu^2]$$ of the resulting Wannier functions. This procedure yields exponentially localized, orthonormal orbitals that transform under crystal symmetry.

The tight-binding Hamiltonian is then formulated in the Wannier basis, with hopping (tunneling) integrals calculated via

$$t_{\nu\nu'}(\mathbf{R}_i) = \langle w_{j+\mathbf{R}_i, \nu} | \hat{H} | w_{j, \nu'} \rangle,$$

yielding either real-space or $k$-space models of the form

$$H(\mathbf{k})_{\nu\nu'} = \sum_{\mathbf{R}_i} t_{\nu\nu'}(\mathbf{R}_i) e^{i\mathbf{k}\cdot\mathbf{R}_i}.$$

This procedure is standard for electronic systems (Jung et al., 2013), optical lattices (Modugno et al., 2015), photonic crystals (Morales-Pérez et al., 2023), and topological materials (Li et al., 2021), with symmetry, topology, and orbital character directly encoded in the resulting MLWFs and hopping matrices.

2. Model Construction: Range, Symmetry, and Significance of Hopping Terms

Wannier tight-binding models systematically include local (onsite) energies and hopping processes between MLWFs at various distances, with the range and symmetry of these processes determined both by the localization of the basis and the topology of the band manifold. For example:

• In monolayer and bilayer graphene, models constructed using MLWFs reveal that both intralayer and interlayer hopping extend significantly beyond nearest neighbors, contradicting earlier phenomenological TB models and capturing features such as particle-hole symmetry breaking and trigonal warping (Jung et al., 2013).
• In twisted bilayer graphene and TMD moiré systems, the minimal number and spatial arrangement of Wannier functions can be dictated by topological obstructions or fragile band topology, requiring composite-band or multi-orbital approaches for an accurate low-energy model (Kang et al., 2018, Cao et al., 2020, Crépel et al., 22 Mar 2024).
• In optical lattices, Wannier TB models enable the mapping from continuum (lattice potential) descriptions to discrete models, where the importance of retaining up to third (or higher) neighbor processes is quantified via explicit calculation of MLWF overlap integrals (1211.6893, Modugno et al., 2015).

The choice and range of hopping terms directly control the model's ability to capture finer details of the band structure, such as indirect gaps or van Hove singularities (e.g., in h-BN (Javvaji et al., 19 Apr 2024)), topological gaps (e.g., in Chern insulator models (Modugno et al., 2015)), or flat band formation at magic twist angles (Carr et al., 2019, Crépel et al., 22 Mar 2024). Analytical expressions fit to ab initio data (e.g., $t_0 = 1.16 s^{0.95} e^{-1.634\sqrt{s}}$ for honeycomb optical lattices) allow experimental parameters to be directly related to microscopic TB coefficients.

3. Symmetry, Topology, and Orbital Character

Wannier tight-binding models capture not only local chemistry but also global symmetries and topological properties of the electronic or photonic structure:

• Models constructed with symmetry-adapted Wannier functions automatically reflect rotation, mirror, inversion, time-reversal, and other space group operations (Carr et al., 2019, Li et al., 2021).
• The orbital character of the MLWFs encodes angular momentum and transformation properties—such as $C_3$ eigenvalues distinguishing s- and p-like orbitals, which underlies topological phase transitions in TMD bilayers (Luo et al., 2022), or chiralities relevant for quantum (anomalous) Hall and spin Hall effects.
• Topological quantum chemistry provides a systematic framework for decomposing the band structure into elementary band representations (EBRs), supporting the model-building for photonic crystals with nontrivial topologies even in the presence of vectorial constraints and $\Gamma$-point singularities (Morales-Pérez et al., 2023).

Where fragile topology or topological obstructions appear (notably in TBG and related moiré systems), models that include all symmetry-allowed orbitals and fill the minimal projective representation space are required to obtain exponentially localized Wannier functions and avoid artificial symmetry breaking (Kang et al., 2018, Cao et al., 2020, Carr et al., 2019).

4. Model Validation, Truncation, and Ab Initio Accuracy

Validation of Wannier-based tight-binding models is carried out by comparing their band structures—and, crucially, wavefunction properties (such as interband velocities, Berry connection, and optical matrix elements)—to the original ab initio data:

• Agreement in eigenvalues is a necessary but not sufficient condition; accurate reproduction of wavefunction properties is essential for predicting optical responses, dielectric functions, shift current, and other physical observables (Ghosh et al., 24 Sep 2024).
• Truncation of hopping terms (e.g., to next-nearest neighbor only) can degrade accuracy, particularly for optical or topological responses, unless the Wannier basis is sufficiently complete and well-localized (Ghosh et al., 24 Sep 2024).
• Symmetry-enforced post-processing, such as group averaging and automated workflows (AiiDA platform), can restore exact degeneracies and facilitate large-scale, high-throughput model construction (Gresch et al., 2018).
• Stitching and hybridization of TB models in different regions (e.g., bulk versus surface or layered versus interface) require special attention to phase, spin, and orbital alignment of overlapping Wannier functions; methods for seamless stitching and Hamiltonian correction are established (Lihm et al., 2019).

Practical implementation relies on open-source codes such as Wannier90, MLWF parametrization, and interfaces with DFT engines. Automated workflows for energy-window optimization, symmetry handling, and parameter interpolation (e.g., under strain) are in place for scaling to complex or strained device geometries (Gresch et al., 2018).

5. Representative Applications Across Material Classes

Wannier tight-binding models have become integral for modeling and interpreting the electronic, optical, and topological properties in a broad array of systems:

System/Class	Key Features Enabled by Wannier TB	References
Monolayer & Bilayer Graphene	Full-range hopping, particle-hole asymmetry,	(Jung et al., 2013)
	trigonal warping, SWM parameter resolution
Twisted Bilayer Graphene	Fragile topology, three-peak Wannier functions,	(Kang et al., 2018, Carr et al., 2019, Cao et al., 2020)
	interplay of topology and interactions
TMD Homobilayer and Heterobilayer	Chern number sequences, magic angle, orbital	(Crépel et al., 22 Mar 2024, Luo et al., 2022)
	interference, topological transitions
Hexagonal Boron Nitride	Indirect band gaps, twisted stacking,	(Javvaji et al., 19 Apr 2024)
	strain-parametrized hopping
Ultracold Atoms/Optical Lattices	Accurate mapping of continuum to lattice,	(1211.6893, Modugno et al., 2015)
	Haldane & Peierls physics, experimental tuning
Bismuth Allotropes	Quantum spin Hall, crystalline insulator phases,	(Li et al., 2021)
	non-symmorphic symmetries, topological invariants
3D Photonic Crystals	$\Gamma$-point obstruction, transversality, TETB	(Morales-Pérez et al., 2023)
Weyl/Dirac Semimetals	Berry curvature, Fermi arcs, field-induced nodes	(Villanova et al., 2018)
Non-Hermitian Systems	Bi-orthogonal Wannier functions, skin effect	(Mochizuki et al., 2022)
Optoelectronic/Nonlinear Response	Optical matrix elements, dielectric/shift current	(Ghosh et al., 24 Sep 2024)
Nanowire Qubit Architectures	Analytic reduction from Schrödinger, qubit gates	(Pomorski, 2021)

6. Extensions: Many-Body Effects, Optical and Electromagnetic Coupling

Beyond non-interacting models, Wannier-based tight-binding models naturally accommodate electron-electron, electron-hole, or bosonic interactions:

• Extended Hubbard models in moiré lattice systems account for on-site, nearest-neighbor, and long-range interactions, with precise values derived from real-space integrals over the MLWFs (Koshino et al., 2018, Cao et al., 2020).
• Many-body studies—such as the emergence of Chern ferromagnetism, van Hove driven antiferromagnetic states, and unconventional superconductivity—leverage the ability to project correlated Hamiltonians onto the Wannier basis (Crépel et al., 22 Mar 2024).
• For quantum light-matter coupling, projection onto Wannier orbitals allows direct derivation of gauge-invariant Peierls phases, dipolar coupling, and emergent electron–polariton bands, with care taken regarding gauge (Coulomb versus dipolar) and gauge-transformation choices in truncated models (Li et al., 2020).

Wannier TB methods also serve as a bridge for incorporating electromagnetic couplings, topological invariants (Berry phases, Chern numbers), and non-Hermitian physics (bi-orthogonal bases for open systems (Mochizuki et al., 2022)).

7. Outlook and Research Directions

Recent developments point towards:

• More automated, robust workflows for generating symmetrized, high-accuracy, and transferable Wannier models at scale, even for device-level structures experiencing strain or external fields (Gresch et al., 2018).
• Model building in complex, topologically nontrivial, or non-Hermitian systems—utilizing EBR decomposition, bi-orthogonal constructions, and mapping of continuum theory onto localized bases even in the presence of obstructions or singularities (Morales-Pérez et al., 2023, Mochizuki et al., 2022).
• Exploiting Wannier TB models for nonlinear, quantum, and optoelectronic responses, with a focus on ensuring both band structure and wavefunction accuracy for observables that depend on velocity, Berry connection, and matrix elements (Ghosh et al., 24 Sep 2024).
• Engineering quantum phases and responses via superlattice modulation or artificial gauge fields, as demonstrated in flat band formation under external periodic potentials (Do et al., 14 Dec 2024).

In conclusion, Wannier tight-binding models represent a quantitative, symmetry-conscious, and highly versatile bridge between first-principles calculations and effective theories. Their centrality in the theoretical and computational toolkit continues to expand as research targets increasingly complex quantum materials and engineered systems.