Bloch Bands, Wannier Functions, and Effective Band Theories

Bloch and Wannier descriptions are Fourier-dual ways to represent the same isolated band subspace. The gauge-invariant object is the projector onto that subspace; individual Bloch phases, orbital embeddings, and Wannier centers require declared conventions. A localization obstruction occurs when no smooth, periodic, symmetry-compatible frame exists, not merely when one numerical gauge looks delocalized.

Required background. Effective lattice Hamiltonians supplies the orbital basis and hopping description; vector and associated bundles supplies the bundle language; bundle connections and curvature supplies gauge transformations of local frames.

Helpful background. Discrete and antiunitary symmetries supplies the sewing relations used for symmetry-compatible frames.

Bloch subspaces and their gauge freedom

For orbitals $|\mathbf R,\alpha\rangle$ in a periodic lattice, translation invariance gives

$$H_{\alpha\beta}(\mathbf k)=\sum_{\mathbf R}t_{\alpha\beta}(\mathbf R)e^{i\mathbf k\cdot\mathbf R}, \qquad H(\mathbf k)|u_{n\mathbf k}\rangle=\varepsilon_n(\mathbf k)|u_{n\mathbf k}\rangle.$$

Suppose $N_{\rm occ}$ bands are separated from the rest by a direct gap $\Delta_{\rm dir}(\mathbf k)>0$ everywhere. Their projector

$$P(\mathbf k)=\sum_{n=1}^{N_{\rm occ}}|u_{n\mathbf k}\rangle\langle u_{n\mathbf k}|$$

is unchanged by a momentum-dependent frame rotation $|u_n\rangle\mapsto|u_m\rangle U_{mn}(\mathbf k)$ with $U\in U(N_{\rm occ})$. Projected observables, Berry curvature, and Wilson-loop eigenvalues must therefore be expressible through $P$ or transform covariantly. Individual eigenvectors are coordinate choices on the occupied bundle Marzari et al. 2012, §§ II–III.

Wannier transformation and localization

Given a smooth periodic frame, a composite Wannier orbital is

$$|\mathbf R,a\rangle=\frac{V_c}{(2\pi)^d}\int_{\rm BZ}d^dk\, e^{-i\mathbf k\cdot\mathbf R}|\psi_{a\mathbf k}\rangle, \qquad |\psi_{a\mathbf k}\rangle=e^{i\mathbf k\cdot\mathbf r}|u_{a\mathbf k}\rangle.$$

Analytic continuation of the frame into a complex neighborhood of the Brillouin torus implies exponential real-space localization. In two and three dimensions, a nonzero first Chern class forbids a globally smooth periodic occupied frame and hence exponentially localized composite Wannier functions for the entire subspace Brouder et al. 2007. A time-reversal topological insulator has zero total Chern class, so localized Wannier functions may exist, but no frame can simultaneously be smooth, periodic, and organized into the desired time-reversal Kramers pairs.

For entangled bands, an outer energy window does not define a unique subspace. One first chooses a smooth rank-$N$ projector inside that window, usually by minimizing a gauge-invariant spillage or spread functional, and then localizes within it. The answer depends on the chosen window and target subspace; a disentanglement algorithm cannot establish a topological obstruction unless those choices and their stability are reported.

Symmetry sewing and effective models

If a symmetry $g$ maps $\mathbf k$ to $g\mathbf k$, its occupied sewing matrix is

$$B_g(\mathbf k)_{mn}=\langle u_{m,g\mathbf k}|\hat g|u_{n\mathbf k}\rangle.$$

It changes covariantly under occupied-frame rotations, while its eigenvalues at symmetry-fixed momenta and compatible products along invariant lines can be gauge invariant. A Wannier tight-binding model is faithful only when it reproduces the target projector, symmetry sewing, and energy window—not merely selected eigenvalues. Boundary calculations additionally require the actual orbital embedding and termination.

Validity limits

The single-particle construction assumes a well-defined isolated subspace. A direct-gap closing invalidates it even if an indirect insulating gap appears elsewhere; strong interactions replace $P(\mathbf k)$ by many-body response, Green-function, entanglement, or defect data. Disorder removes crystal momentum, although a mobility-gapped real-space projector can retain topological information. Wannier centers are gauge and unit-cell dependent; only their symmetry-quantized combinations or changes along a gapped path are physical.

Exercise

Show that the projector is invariant under an occupied-frame rotation $U(\mathbf k)$.

Solution

Writing $|u'_a\rangle=|u_n\rangle U_{na}$ gives $P'=\sum_a|u_n\rangle U_{na}U^*_{ma}\langle u_m|=\sum_{nm}|u_n\rangle(UU^\dagger)_{nm}\langle u_m|=P$. Thus a formula built only from $P$ cannot depend on the chosen occupied frame.

References

• Christian Brouder, Gianluca Panati, Matteo Calandra, Christophe Mourougane, and Nicola Marzari, “Exponential Localization of Wannier Functions in Insulators,” Physical Review Letters 98 (2007) 046402, doi:10.1103/PhysRevLett.98.046402.
• Nicola Marzari, Arash A. Mostofi, Jonathan R. Yates, Ivo Souza, and David Vanderbilt, “Maximally Localized Wannier Functions: Theory and Applications,” Reviews of Modern Physics 84 (2012) 1419–1475, doi:10.1103/RevModPhys.84.1419.