Localized Non-Bloch Wannier Functions

• Localized non-Bloch Wannier functions are real-space bases that generalize traditional Bloch constructions to non-periodic systems and perturbative scenarios.
• They utilize methods like projector-based algorithms, Löwdin orthogonalization, and phase-space strategies to achieve exponential localization and overcome conventional limitations.
• Recent developments apply these functions to accurate band interpolation, response function analysis, and managing topological obstructions in complex materials.

Searching arXiv for the cited papers and related terminology. Searching arXiv for “Wannier function perturbation theory” and generalized/non-periodic Wannier constructions. Localized non-Bloch Wannier functions are localized real-space bases or Wannier-like objects that extend the usual Wannier construction beyond a finite set of Bloch eigenstates of a periodic Hermitian Hamiltonian. The expression is not used uniformly across the literature. In the works surveyed here, it covers at least five related constructions: exponentially localized generalized Wannier bases for non-periodic insulators obtained from the Fermi projector, compactly supported bases associated with strictly local projectors, projection-and-Löwdin schemes built from localized trial orbitals or guiding functions, phase-space Wannier bases that do not presuppose a crystal Bloch decomposition, and localized perturbative objects such as Wannier function perturbations. The common structural feature is a localized representation of a spectral or response subspace when ordinary Bloch-band Wannierization is unavailable, insufficient, or deliberately bypassed (Lu et al., 2020, Stubbs et al., 2020, Zhu et al., 2016, Lihm et al., 2021).

1. Conventional Wannier theory and the origin of the non-Bloch extension

In the conventional formulation, Wannier functions are obtained from Bloch eigenstates of a periodic Hamiltonian by a momentum-space unitary gauge choice. For a set of $J$ bands, one writes

$$|{\bf R} n \rangle = \frac{V}{(2\pi)^3}\int d{\bf k}\, e^{-i{\bf k}\cdot {\bf R}}|\tilde \psi_{n \bf k}\rangle,$$

with $$|\tilde\psi_{n\mathbf k}\rangle = \sum_{m=1}^J U^{\mathbf k}_{mn}|\psi_{m\mathbf k}\rangle$$. Localization is therefore a gauge problem inside a finite Bloch-band manifold, and standard MLWF theory minimizes the quadratic spread over the unitary matrices $U^{\mathbf k}$ (Wang et al., 2014).

Two distinct limitations motivate the non-Bloch generalization. First, many physically important quantities depend on sums over all bands rather than on a truncated low-energy manifold. In particular, wavefunction perturbations

$$\ket{\psi^{(1)}_{n}}=\sum_{n'}' \ket{\psi^{(0)}_{n'}} \frac{\langle \psi^{(0)}_{n'}|V|\psi^{(0)}_{n}\rangle}{\varepsilon^{(0)}_n-\varepsilon^{(0)}_{n'}}$$

cannot be represented accurately inside a finite Wannier basis (Lihm et al., 2021). Second, in non-crystalline systems Bloch momentum and the Brillouin zone do not exist, so the usual Fourier-transform definition of Wannier functions is unavailable; the relevant object becomes the Fermi projector and its localized real-space structure (Lu et al., 2020). This suggests that “non-Bloch” should be understood functionally: it designates localized representatives of spectral or response subspaces that are not exhausted by the standard Fourier transform of a finite Bloch manifold.

2. Projector-based generalized Wannier functions in non-periodic systems

A fully non-Bloch construction begins with the Fermi projection

$P=\chi_{(-\infty,E_F)}(H),$

or, more generally, the spectral projection onto an isolated part of the spectrum. For insulating but not necessarily crystalline systems, exponentially localized generalized Wannier functions are defined as an orthonormal basis of $\mathrm{range}(P)$ with uniform exponential decay away from their centers. In two dimensions and higher, one rigorous route is to study projected position operators such as $PXP$ and $P_jYP_j$, where $P_j$ are spectral projectors of $$|{\bf R} n \rangle = \frac{V}{(2\pi)^3}\int d{\bf k}\, e^{-i{\bf k}\cdot {\bf R}}|\tilde \psi_{n \bf k}\rangle,$$0. Under the assumption that $$|{\bf R} n \rangle = \frac{V}{(2\pi)^3}\int d{\bf k}\, e^{-i{\bf k}\cdot {\bf R}}|\tilde \psi_{n \bf k}\rangle,$$1 decomposes into uniformly separated compact spectral sets of uniformly bounded width, the eigenfunctions of a sequence of such self-adjoint projected-position operators form an exponentially localized orthonormal basis of $$|{\bf R} n \rangle = \frac{V}{(2\pi)^3}\int d{\bf k}\, e^{-i{\bf k}\cdot {\bf R}}|\tilde \psi_{n \bf k}\rangle,$$2 (Lu et al., 2020).

The corresponding numerical realization is the Iterated Projected Position (IPP) algorithm. It replaces gauge optimization by a sequence of matrix diagonalizations: first diagonalize $$|{\bf R} n \rangle = \frac{V}{(2\pi)^3}\int d{\bf k}\, e^{-i{\bf k}\cdot {\bf R}}|\tilde \psi_{n \bf k}\rangle,$$3, then diagonalize $$|{\bf R} n \rangle = \frac{V}{(2\pi)^3}\int d{\bf k}\, e^{-i{\bf k}\cdot {\bf R}}|\tilde \psi_{n \bf k}\rangle,$$4 inside each spectral slice, and iterate in higher dimensions. The method applies to both periodic and non-periodic systems, requires no initialization, cannot get stuck at a local minimum, and is supported by a rigorous analysis under mild assumptions. It is demonstrated for the Haldane model, the Kane–Mele model in both $$|{\bf R} n \rangle = \frac{V}{(2\pi)^3}\int d{\bf k}\, e^{-i{\bf k}\cdot {\bf R}}|\tilde \psi_{n \bf k}\rangle,$$5-even and $$|{\bf R} n \rangle = \frac{V}{(2\pi)^3}\int d{\bf k}\, e^{-i{\bf k}\cdot {\bf R}}|\tilde \psi_{n \bf k}\rangle,$$6-odd phases, and the $$|{\bf R} n \rangle = \frac{V}{(2\pi)^3}\int d{\bf k}\, e^{-i{\bf k}\cdot {\bf R}}|\tilde \psi_{n \bf k}\rangle,$$7 model on a quasi-crystal lattice (Stubbs et al., 2020).

In this projector-based line of work, the non-Bloch character is literal rather than metaphorical. The localized basis is constructed directly from $$|{\bf R} n \rangle = \frac{V}{(2\pi)^3}\int d{\bf k}\, e^{-i{\bf k}\cdot {\bf R}}|\tilde \psi_{n \bf k}\rangle,$$8 and position operators, not from Bloch states. In periodic systems the resulting basis reduces to ordinary exponentially localized Wannier functions when topological obstructions vanish; in non-periodic systems it remains well defined even though Bloch theory is absent (Lu et al., 2020, Stubbs et al., 2020).

3. Projection, Löwdin orthogonalization, and guiding-orbital constructions

A second family of constructions keeps locality primary and treats Bloch information as a projection target rather than as the starting point. One algorithm proceeds in three steps: choose localized trial functions $$|{\bf R} n \rangle = \frac{V}{(2\pi)^3}\int d{\bf k}\, e^{-i{\bf k}\cdot {\bf R}}|\tilde \psi_{n \bf k}\rangle,$$9, project them with the full band projector $$|\tilde\psi_{n\mathbf k}\rangle = \sum_{m=1}^J U^{\mathbf k}_{mn}|\psi_{m\mathbf k}\rangle$$0, and orthonormalize the projected set by the Löwdin method. Because the projection is over the entire isolated band subspace, this framework can be applied to random potentials without supercells and without requiring the eigenfunctions to be Bloch waves. The resulting Wannier functions are obtained as the orthonormalized set closest in least-squares sense to the projected localized trials (Zhu et al., 2016).

A closely related but explicitly Bloch-subspace formulation is the construction of closest Wannier functions. Given localized guiding functions $$|\tilde\psi_{n\mathbf k}\rangle = \sum_{m=1}^J U^{\mathbf k}_{mn}|\psi_{m\mathbf k}\rangle$$1, one projects them into a windowed Bloch subspace,

$$|\tilde\psi_{n\mathbf k}\rangle = \sum_{m=1}^J U^{\mathbf k}_{mn}|\psi_{m\mathbf k}\rangle$$2

with $$|\tilde\psi_{n\mathbf k}\rangle = \sum_{m=1}^J U^{\mathbf k}_{mn}|\psi_{m\mathbf k}\rangle$$3, and then minimizes the distance measure

$$|\tilde\psi_{n\mathbf k}\rangle = \sum_{m=1}^J U^{\mathbf k}_{mn}|\psi_{m\mathbf k}\rangle$$4

under the orthogonality constraint $$|\tilde\psi_{n\mathbf k}\rangle = \sum_{m=1}^J U^{\mathbf k}_{mn}|\psi_{m\mathbf k}\rangle$$5. The minimizer is the partial-unitary polar factor $$|\tilde\psi_{n\mathbf k}\rangle = \sum_{m=1}^J U^{\mathbf k}_{mn}|\psi_{m\mathbf k}\rangle$$6 from the SVD of $$|\tilde\psi_{n\mathbf k}\rangle = \sum_{m=1}^J U^{\mathbf k}_{mn}|\psi_{m\mathbf k}\rangle$$7, so the gauge is fixed non-iteratively by polar decomposition. This accommodates atomic orbitals, hybrid orbitals, and embedded molecular orbitals as guides, and it addresses disentanglement through a smoothly varying window function rather than through spread minimization (Ozaki, 2023).

Two further constructions enlarge the notion of a Wannier basis beyond crystal bands. In “Quantum phase space with a basis of Wannier functions,” a complete orthonormal basis $$|\tilde\psi_{n\mathbf k}\rangle = \sum_{m=1}^J U^{\mathbf k}_{mn}|\psi_{m\mathbf k}\rangle$$8 is built from a von Neumann lattice of Gaussian packets by combining Kohn’s method with Löwdin orthogonalization. Each basis function is localized at a Planck cell of area $$|\tilde\psi_{n\mathbf k}\rangle = \sum_{m=1}^J U^{\mathbf k}_{mn}|\psi_{m\mathbf k}\rangle$$9, the mapping $U^{\mathbf k}$0 is unitary, and the basis is not derived from eigenstates of any periodic Hamiltonian (Fang et al., 2017). In “Position scaling-eigenfunctions,” one first constructs real-space localized eigenvectors of a projected position operator inside a wavelet scaling space and only afterward projects them onto a Bloch-band subspace, obtaining PPEs and PWEs that accurately approximate MLWFs. Here the non-Bloch localized basis precedes the band projection rather than resulting from it (Hamai et al., 2023).

4. Perturbative and response-space Wannier-like functions

A different extension moves beyond Bloch eigenstates by localizing response directions in Hilbert space. Wannier Function Perturbation Theory introduces Wannier function perturbations (WFPs),

$U^{\mathbf k}$1

defined as first-order changes of Wannier functions under a perturbation $U^{\mathbf k}$2. These objects are not unperturbed Bloch eigenstates and are not ordinary Wannier functions of a fixed Hamiltonian; rather, they are localized representatives of wavefunction perturbations that incorporate contributions from infinitely many bands (Lihm et al., 2021).

The formalism organizes the Hilbert space through a perturbation window $U^{\mathbf k}$3, an inner or frozen window $U^{\mathbf k}$4, and an outer disentanglement window $U^{\mathbf k}$5, with $U^{\mathbf k}$6. The central WFP expression combines Sternheimer-computed wavefunction perturbations with a correction involving $U^{\mathbf k}$7, thereby recovering contributions from states outside the Wannier subspace while avoiding singular denominators. The paper gives two localization arguments: continuity of projection-only Wannierization under small perturbation, and smoothness of each $U^{\mathbf k}$8-space summand when the $U^{\mathbf k}$9 restriction is maintained (Lihm et al., 2021).

The significance is that quantities usually regarded as irreducibly cross-band can be recast into localized real-space objects. The paper demonstrates this for temperature-dependent indirect optical absorption in silicon, shift spin conductivity in monolayer WTe$$\ket{\psi^{(1)}_{n}}=\sum_{n'}' \ket{\psi^{(0)}_{n'}} \frac{\langle \psi^{(0)}_{n'}|V|\psi^{(0)}_{n}\rangle}{\varepsilon^{(0)}_n-\varepsilon^{(0)}_{n'}}$$0, and spin Hall conductivity of the same material, all without band-truncation error. It explicitly notes that the phrase “localized non-Bloch Wannier functions” is not used and that non-Hermitian or genuinely non-Bloch band theory is not treated, but conceptually the construction is very close: locality is transferred from eigenstates to their perturbative directions (Lihm et al., 2021).

5. Compact support, gauge, and topological obstruction

Compact support is a substantially stronger property than exponential localization. In one-dimensional non-interacting tight-binding models, a subspace possesses a compactly supported orthogonal basis if and only if the orthogonal projector onto that subspace is strictly local, meaning

$$\ket{\psi^{(1)}_{n}}=\sum_{n'}' \ket{\psi^{(0)}_{n'}} \frac{\langle \psi^{(0)}_{n'}|V|\psi^{(0)}_{n}\rangle}{\varepsilon^{(0)}_n-\varepsilon^{(0)}_{n'}}$$1

This criterion is independent of lattice translation symmetry. For higher dimensions, the paper gives additional sufficient conditions, notably nearest-neighbor reducibility, and proves that a projector is strictly local if and only if for any chosen axis its image is spanned by hybrid Wannier functions compactly supported along that axis (Sathe et al., 2020).

Topology sharply constrains how far localization can be pushed. A homotopy-based algorithm constructs localized Bloch-based Wannier functions whenever the system has vanishing Chern numbers; it works in topological insulators such as the Kane–Mele model because a non-symmetric smooth Bloch frame exists even though a time-reversal-symmetric one does not (Gontier et al., 2018). By contrast, compressed-sensing minimization over entire topological equivalence classes shows that in one-dimensional class-D topological superconductors one can flow to a representative with compact Wannier functions supported on two sites, while in two-dimensional class-AII topological insulators the search yields numerical evidence for the absence of compact Wannier functions. In that setting, compact support would imply an exactly flat-band model with strictly finite-range hopping, and the algorithm does not find one (Budich et al., 2014).

These results correct two common misconceptions. First, non-Bloch localization does not eliminate topological obstruction; it merely changes the language in which the obstruction is expressed. Second, vanishing Chern numbers are sufficient for exponentially localized Wannier functions in periodic systems, but not for compactly supported ones. This suggests that compact non-Bloch Wannier bases are exceptional objects tied to strictly local projectors or to special flat-band representatives, not generic consequences of trivial topology (Sathe et al., 2020, Gontier et al., 2018, Budich et al., 2014).

6. Applications, selective localization, and current direction of the field

Several recent developments show that localized non-Bloch Wannier constructions are not only existence results but also workflow components in electronic-structure practice. Selectively localized Wannier functions generalize MLWFs by localizing only a subset of orbitals of interest, while optionally fixing their centers and preserving point-group symmetry. This is explicitly motivated by beyond-DFT methods in which only a correlated local subspace must be especially localized and symmetry adapted (Wang et al., 2014).

A complementary post-processing strategy starts from one large MLWF set spanning several manifolds simultaneously and remixes it into orthogonal localized sets for target submanifolds. The procedure uses the Wannier-gauge Hamiltonian, energy-based partitioning, parallel transport, and final submanifold localization. It is demonstrated for the valence and conduction bands of silicon, the top valence band of MoS$$\ket{\psi^{(1)}_{n}}=\sum_{n'}' \ket{\psi^{(0)}_{n'}} \frac{\langle \psi^{(0)}_{n'}|V|\psi^{(0)}_{n}\rangle}{\varepsilon^{(0)}_n-\varepsilon^{(0)}_{n'}}$$2, the $$\ket{\psi^{(1)}_{n}}=\sum_{n'}' \ket{\psi^{(0)}_{n'}} \frac{\langle \psi^{(0)}_{n'}|V|\psi^{(0)}_{n}\rangle}{\varepsilon^{(0)}_n-\varepsilon^{(0)}_{n'}}$$3 and $$\ket{\psi^{(1)}_{n}}=\sum_{n'}' \ket{\psi^{(0)}_{n'}} \frac{\langle \psi^{(0)}_{n'}|V|\psi^{(0)}_{n}\rangle}{\varepsilon^{(0)}_n-\varepsilon^{(0)}_{n'}}$$4 bands of SrVO$$\ket{\psi^{(1)}_{n}}=\sum_{n'}' \ket{\psi^{(0)}_{n'}} \frac{\langle \psi^{(0)}_{n'}|V|\psi^{(0)}_{n}\rangle}{\varepsilon^{(0)}_n-\varepsilon^{(0)}_{n'}}$$5, and a mid-throughput study of 77 insulators (Qiao et al., 2023).

The broader application space is wider than band interpolation. Closest Wannier functions reproduce targeted conventional bands in Si, Cu, the TTF-TCNQ molecular crystal, and Bi$$\ket{\psi^{(1)}_{n}}=\sum_{n'}' \ket{\psi^{(0)}_{n'}} \frac{\langle \psi^{(0)}_{n'}|V|\psi^{(0)}_{n}\rangle}{\varepsilon^{(0)}_n-\varepsilon^{(0)}_{n'}}$$6Se$$\ket{\psi^{(1)}_{n}}=\sum_{n'}' \ket{\psi^{(0)}_{n'}} \frac{\langle \psi^{(0)}_{n'}|V|\psi^{(0)}_{n}\rangle}{\varepsilon^{(0)}_n-\varepsilon^{(0)}_{n'}}$$7, and they are also used to define effective atomic charges (Ozaki, 2023). Wannier function perturbation theory extends localized interpolation to indirect optical absorption, shift spin conductivity, and spin Hall conductivity without band-truncation error (Lihm et al., 2021). The phase-space Wannier basis provides a unitary true-probability representation in quantum phase space and is proposed for time-frequency analysis of signals (Fang et al., 2017).

Taken together, these developments indicate a shift in emphasis. The central object is no longer always the Bloch eigenstate manifold itself, but rather a localized representation of whatever subspace is physically operative: occupied states, target manifolds, projected trial-orbital images, compact projector ranges, phase-space cells, or perturbative response directions. That is the precise sense in which localized non-Bloch Wannier functions have emerged as a unifying concept across modern Wannier theory.