What is disentanglement?

Band entanglement occurs when the electronic states that we want to describe overlap in energy, cross, or hybridize with other bands. In this situation, the target states can no longer be identified simply by following the same band indices throughout the Brillouin zone. Disentanglement is the procedure used to extract the desired smoothly connected electronic subspace from this larger, mixed set of Bloch states.

To understand disentanglement, imagine that we want to follow a particular group of electronic states through the Brillouin zone.

For an isolated group of bands, this is simple. The target bands are separated from all other bands by energy gaps, so the same group can be followed continuously from one k-point to another.

For entangled bands, however, the situation is different. The states of interest cross and hybridize with other states. As a result, a state with a particular orbital character may correspond to one DFT band index at one k-point and to another band index at a neighboring k-point.

Therefore, simply choosing bands by their band numbers does not work.

The key idea of disentanglement is to follow the physical character of the electronic subspace through neighboring k-points.

But what do we mean by the physical character of an electronic state?

A Bloch state is not characterized only by its energy. It also has a particular wave-function structure. For example, a state may be predominantly localized on a particular atom, may have mainly s-, p-, or d-orbital character, may resemble a particular atomic orbital such as $d_{z^2}$ or $d_{xy}$, or may represent a particular bonding pattern between neighboring atoms. In spin-polarized or spinor calculations, spin character may also be important.

For example, in monolayer MoS₂, the states that we want to describe may have predominantly $d_{z^2}, d_{xy}, d_{x^2-y^2}$ character on the Mo atoms.

As we move from one k-point to another, the energies of these states change, and the corresponding DFT bands may cross or hybridize with other bands. However, their wave functions usually evolve continuously in k-space. A state at one k-point therefore tends to have a strong wave-function connection with the corresponding electronic subspace at nearby k-points.

This is the information used in the disentanglement procedure.

Wannier90 receives two particularly important types of information from the electronic-structure calculation.

First, the projection matrices

$$A_{mn}^{(\mathbf{k})} = \langle
\psi_{m\mathbf{k}} | g_n \rangle$$

measure the overlap between the DFT Bloch states $\psi_{m\mathbf{k}}$ and the localized trial orbitals $g_n$ specified in the projections block.

For example,

begin projections
Mo:dz2
Mo:dxy
Mo:dx2-y2
end projections

defines trial functions with the desired Mo d-orbital character.

If a particular Bloch state has a large overlap with one of these trial orbitals, this indicates that the state contains a significant component of the desired orbital character. These projections provide the initial guess for the Wannier subspace.

Second, Wannier90 uses the overlap matrices between Bloch states at neighboring k-points,

$$M_{mn}^{(\mathbf{k},\mathbf{b})} = \langle u_{m\mathbf{k}} | u_{n,\mathbf{k+b}} \rangle$$

where $u_{n\mathbf{k}}$ is the cell-periodic part of a Bloch function and $\mathbf{b}$ connects neighboring points of the k-point mesh.

These overlaps tell Wannier90 how strongly the electronic states at one k-point are connected to states at a neighboring k-point.

For example, suppose that at $\mathbf{k}_1$ the desired state is mainly represented by DFT band 7. At a neighboring point $\mathbf{k}_2$, because of a band crossing, the continuation of the same wave-function character may be found mainly in band 8 rather than band 7.

Then the relevant connection may schematically be

$$\psi_{7,\mathbf{k}1}
\longrightarrow
\psi_{8,\mathbf{k}_2}$$

rather than

$$\psi_{7,\mathbf{k}1}
\longrightarrow
\psi_{7,\mathbf{k}_2}.
$$

In a more general situation, especially when bands strongly hybridize, the desired state may not correspond to any single DFT band. Its character may be distributed among several Bloch eigenstates. Wannier90 can therefore construct the target subspace from linear combinations of the available states,

$$|\widetilde{\psi}_{n\mathbf{k}}\rangle = \sum_{m=1}^{N_{\mathrm{win}}} |\psi_{m\mathbf{k}}\rangle U^{\mathrm{dis}}_{mn}(\mathbf{k})$$

The disentanglement algorithm searches for the $N_{\mathrm W}$-dimensional subspace that changes as smoothly as possible between neighboring k-points. In practice, this smoothness is determined from the overlaps between neighboring subspaces.

Therefore, the phrase follow the physical character should not be understood as Wannier90 explicitly labeling every band as, for example, “Mo $d_{z^2}$” or “S $p_x$.” Instead:

the projections provide an initial connection to the desired atomic or orbital character;
the neighboring-k-point overlaps show how the wave-function subspace evolves through reciprocal space;
the disentanglement algorithm selects the smoothly connected subspace from the larger set of Bloch states available inside the outer energy window.

In this way, Wannier90 can follow the same physical electronic manifold even when its orbital character moves between different DFT band indices or becomes distributed among several hybridized states.

Consider a simple example. Suppose that we want to construct three Wannier functions describing states with predominantly $d_{z^2}, d_{xy}, d_{x^2-y^2}$ character.

At one k-point, these states might correspond mainly to DFT bands 5, 6, and 7. At another k-point, because of crossings and hybridization, the same physical character may be distributed among bands 6, 7, 8, and 9.

Wannier90 should therefore not blindly follow the band indices:

$$
(5,6,7)_{\mathbf{k}1}
;\not\rightarrow;
(5,6,7)_{\mathbf{k}_2}.
$$

Instead, it examines how the Bloch states at neighboring k-points overlap with one another and searches for an $N_{\mathrm W}$-dimensional subspace that varies as smoothly as possible throughout the Brillouin zone.

Schematically,

$$
\text{many available Bloch states}
\rightarrow
\text{compare neighboring k-points}
\rightarrow
\text{follow a smoothly connected subspace}
$$

For example, if $N_{\mathrm W}=3$, but five Bloch states lie inside the outer energy window, Wannier90 does not necessarily select three individual bands with fixed indices. Instead, at every k-point, the target three-dimensional subspace may be constructed from linear combinations of the available Bloch states,

$$
|\widetilde{\psi}_{n\mathbf{k}}\rangle = \sum_{m=1}^{N_{\mathrm{win}}}
|\psi_{m\mathbf{k}}\rangle U^{\mathrm{dis}}_{mn}(\mathbf{k}).
$$

The coefficients can change with $\mathbf{k}$. This allows the selected subspace to preserve its physical character even when the original DFT bands cross or strongly hybridize.

The algorithm compares neighboring k-points and finds the sequence of subspaces that is most smoothly connected across the Brillouin zone.

This is the central idea of disentanglement:

Disentanglement does not simply select bands according to their energies or band indices. It extracts a smoothly connected electronic subspace by following the wave-function character across neighboring k-points.

The initial projections help Wannier90 identify the type of states that we want to describe. For example,

begin projections
Mo:dz2
Mo:dxy
Mo:dx2-y2
end projections

provides an initial connection between the Bloch states and the desired Mo d-like Wannier functions.

The energy windows then control which states can participate in the disentanglement:

the outer window defines the complete pool of Bloch states that Wannier90 is allowed to use;
the inner or frozen window defines states that must remain inside the selected Wannier subspace.

After a smoothly connected subspace has been constructed, Wannier90 performs the localization procedure.

Therefore, the two procedures solve different problems:

$$\text{Disentanglement: Which electronic subspace should be followed?}$$

$$\text{Localization: Which basis inside this subspace gives the most localized Wannier functions?}$$

The complete workflow is therefore

$$
\text{Bloch states}
\rightarrow
\text{disentanglement}
\rightarrow
\text{smooth electronic subspace}
\rightarrow
\text{localization}
\rightarrow
\text{MLWFs}
$$

This is why disentanglement is essential when the target bands cross or hybridize with other bands: it allows Wannier90 to follow their physical character rather than their numerical band indices.