From a First-Principles Hamiltonian to a Compact Wannier Tight-Binding Model

Let us imagine that we have performed a first-principles calculation of a crystalline material using Quantum ESPRESSO. As a result, we obtain the electronic eigenvalues and eigenstates of the Kohn–Sham Hamiltonian,

$${\mathbf{H}}^{\mathrm{DFT}}|\psi_{n\mathbf{k}}\rangle=\varepsilon_{n\mathbf{k}}|\psi_{n\mathbf{k}}\rangle$$

Suppose that, for each $\mathbf{k}$-point, we compute 50 occupied bands and 50 unoccupied bands. In this case, the electronic structure is represented in a space of 100 Bloch states,

$$|\psi_{1\mathbf{k}}\rangle,|\psi_{2\mathbf{k}}\rangle,\dots,|\psi_{100\mathbf{k}}\rangle$$

In this basis, the Hamiltonian is formally a $100 \times 100$ matrix,

$$H^{\mathrm{Bloch}}_{mn}(\mathbf{k})=\langle\psi_{m\mathbf{k}}|{\mathbf{H}}^{\mathrm{DFT}}| \psi_{n\mathbf{k}}\rangle$$

However, in many practical problems we do not need all 100 bands. Very often, the most important physics is controlled only by a small number of bands near the Fermi level: a few valence bands close to the valence-band maximum (VBM) and a few conduction bands close to the conduction-band minimum (CBM).

A good example is monolayer MoS2. Near the band gap, the relevant electronic states are mainly formed by molybdenum d-orbitals. In the simplest model, one can use only three Mo d-orbitals,

$$d_{z^2}, \qquad d_{xy}, \qquad d_{x^2-y^2}$$

These orbitals are sufficient to reproduce the main low-energy bands near the K and K’ valleys. Therefore, instead of working with a large $100 \times 100$ Hamiltonian, one can build a compact $3 \times 3$ tight-binding model,

$$H^{\mathrm{Wannier}}_{3 \times 3}(\mathbf{k})$$

For a more accurate description of a wider energy window, one may include additional orbitals. A commonly used model for MoS2 contains 11 orbitals: five Mo d-orbitals and three p-orbitals from each of the two sulfur atoms,

$$5 \times d_{\mathrm{Mo}}+3 \times p_{\mathrm{S}_1}
+
3 \times p_{\mathrm{S}_2} = 11.$$

This gives an $11 \times 11$ Wannier Hamiltonian,

$$H^{\mathrm{Wannier}}_{11 \times 11}(\mathbf{k})$$

which can describe not only the band-edge states, but also a larger part of the valence and conduction band structure.

This is exactly where Wannier90 becomes useful. Starting from the Bloch states obtained from Quantum ESPRESSO, Wannier90 constructs localized Wannier functions,

$$|w_{n\mathbf{R}}\rangle=\frac{1}{\sqrt{N}}\sum_{\mathbf{k}}e^{-i\mathbf{k}\cdot\mathbf{R}}|\psi_{n\mathbf{k}}\rangle$$

and transforms the electronic Hamiltonian from the delocalized Bloch basis to a localized Wannier basis,

$$H^{\mathrm{Bloch}}_{100 \times 100}

 \quad \longrightarrow
 
\quad H^{\mathrm{Wannier}}_{3 \times 3}
\quad\text{or}\quad H^{\mathrm{Wannier}}_{11 \times 11}$$

In other words, Wannier90 allows us to replace a large first-principles Hamiltonian by a much smaller effective tight-binding Hamiltonian that keeps only the physically important degrees of freedom.

Once this Wannier Hamiltonian is constructed, it can be used for many further calculations. For example, one can:

$$H^{\mathrm{Wannier}}(\mathbf{R})\quad \Rightarrow \quad H^{\mathrm{Wannier}}(\mathbf{k})$$

and then compute electronic band structures on arbitrary high-resolution $\mathbf{k}$-paths, densities of states, Fermi surfaces, effective masses, Berry curvature, topological properties, optical matrix elements, and transport-related quantities.

Thus, the main idea of Wannier90 is not simply to produce localized orbitals. Its deeper purpose is to build a compact, accurate, and physically transparent Hamiltonian that connects first-principles calculations with intuitive tight-binding models.

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