wannier method basics

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docs/hands-on/wannier.mdx

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+---
+title: Wannier method
+---
+
+## Obtain bandstructure of Silicon
+
+1. Perform `scf` calculation using Quantum Espresso `pw.x`
+
+```console
+QE_PATH="/workspaces/q-e-qe-7.2/bin"
+mpirun -np 4 ${QE_PATH}/pw.x -i pw.scf.silicon.in > pw.scf.silicon.out
+```
+
+2. Perform `nscf` calculation using `pw.x`. Instead of automatic k-grid, we need
+to provide explicit list of k-points. Such explicit list of k-points can be
+generated using perl script included in the Wannier package under utility.
+
+```console
+WANNIER_PATH="/workspaces/wannier90-3.1.0"
+# directly append the k-points to the input file
+${WANNIER_PATH}/utility/kmesh.pl 4 4 4 >> pw.nscf.silicon.in
+```
+
+Run `nscf` calculation:
+
+```console
+mpirun -np 4 ${QE_PATH}/pw.x -i pw.nscf.silicon.in > pw.nscf.silicon.out
+```
+
+3. Prepare input file for wannier90 (`silicon.win`). Here we need the k-points
+list without the weights:
+
+```console
+${WANNIER_PATH}/utility/kmesh.pl 4 4 4 wan
+```
+
+4. Generate nnkp input:
+
+```console
+# we can just provide the seedname or seedname.win
+${WANNIER_PATH}/wannier90.x -pp silicon
+```
+
+5. Create input file for `pw2wan`, and generate initial projections:
+
+```console
+mpirun -np 4 ${WANNIER_PATH}/pw2wannier90.x -i pw2wan.silicon.in > pw2
+wan.silicon.out
+```
+
+6. Run wannier calculation:
+
+```console
+mpirun -np 4 ${WANNIER_PATH}/wannier90.x silicon
+```
+
+<picture>
+  <source type="image/webp" srcSet={require("/img/silicon-wannier-bands.webp").default} />
+  <img src={require("/img/silicon-wannier-bands.png").default} alt="silicon-wannier-bands" />
+</picture>
+
+## Resources
+
+- <https://sites.google.com/view/hubbard-koopmans/program>

docs/theory/wannier.md

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+---
+title: Wannier basis
+---
+
+## Introduction
+
+Wannier functions are an alternative representation of Bloch states in terms of
+a localized basis set. Suppose we have $N$ atoms each separated by lattice
+constant $a$ in one dimension. Bloch states are indexed by the wave vector $k$.
+
+$$
+\mathcal{H}\ket{k} = \epsilon_k \ket{k}
+$$
+
+$$
+\braket{x | k} = \psi_k(x)
+$$
+
+From the Bloch theorem, we have
+
+$$
+\psi_k(x + a) = e^{ika}\psi_k(x)
+$$
+
+Now, we want to find an alternative representation in terms of Wannier basis
+$\ket{n}$, where the states are labeled using site index (n = 1, 2, ..., N)
+instead of quantum number $k$. Wannier basis set is complete and orthonormal.
+
+$$
+\ket{k} = \sum_{n=1}^N a_{nk} \ket{n}
+$$
+
+$$
+\braket{x | k} = \sum_n a_{nk} \braket{x | n}
+$$
+
+$$
+\Rightarrow \psi_k(x) = \sum_n a_{nk} w(x - na)
+$$
+
+where $w$ represents Wannier function. Apply translation operator on both sides:
+
+$$
+\begin{aligned}
+&T_a \psi_k(x) = \sum_n a_{nk} T_a w(x - na) \\
+
+\Rightarrow~ & \psi_k(x + a) = \sum_n a_{nk} w\bigl(x - (n-1)a\bigr) \\
+
+\Rightarrow~ & e^{ika} \psi_k(x) = \sum_n a_{(n+1)k} w(x + na) \\
+
+\Rightarrow~ & e^{ika} \sum_n a_{nk} w(x - na) = \sum_n a_{(n+1)k} w(x + na) \\
+
+\Rightarrow~ & e^{ika} \sum_n a_{nk} w(x - na) = \sum_n a_{(n+1)k} w(x + na) \\
+\end{aligned}
+$$
+
+$$
+\begin{aligned}
+\therefore ~ a_{(n+1)k} &= a_{nk} e^{ika} \\
+
+&= a_{(n-1)k}e^{i2ka} \\
+&\qquad\cdots \\
+&= a_{0 k}e^{i(n+1)ka}
+\end{aligned}
+$$
+
+Since Bloch states are orthonormal:
+
+$$
+\begin{aligned}
+\braket{k | k} &= \braket{k | \sum_n a_{nk} | n} \\
+&= \sum_{mn} \braket{m | a_{mk}^* a_{nk} | n} \\
+&= \sum_{n} | a_{nk} |^2 \\
+&= N a_{0k}^2 \\
+&= 1
+\end{aligned}
+$$
+
+$$
+\Rightarrow a_{0k} = \frac{1}{\sqrt{N}} \qquad\text{(up to a phase factor)}
+$$
+
+$$
+\ket{k} = \frac{1}{\sqrt{N}} \sum_n e^{ikna} \ket{n}
+$$
+
+While Bloch states represent the eigenstates of the single-particle Hamiltonian,
+WF (in general) cannot be assigned a single eigen-value, instead WFs are
+obtained as liner combination of Bloch states corresponding to different
+energies.
+
+The Hamiltonian can now be written as:
+
+$$
+\begin{aligned}
+\mathcal{H} &= \sum_m \sum_n \ket{m} \braket{m | \mathcal{H} | n} \bra{n} \\
+&= \sum_n \epsilon_n \ket{n} \bra{n} + \sum_{m\neq n} (-t_{mn}) \ket{m} \bra{n}
+\end{aligned}
+$$
+
+where $\epsilon_n$ is onsite or diagonal term and $t_{mn}$ (> 0) is hopping or
+off-diagonal term.
+
+## Maximally Localized Wannier Function
+
+The choice of Wannier function is not unique. One such option could be the set
+that maximizes localization. Two different sets of Wannier basis are connected
+via unitary transformation. MLWFs can be considered as a generalization of
+localized molecular orbitals (LMOs) to periodic systems.
+
+## Resources
+
+- [Introduction to Wannier Basis lecture by Vijay A. Singh](https://youtu.be/_XWIwoE7Pc4)
+- [*Maximally localized generalized Wannier functions for composite energy bands*, Marzari and Vanderbilt, Phys. Rev. B **56**, 12847 (1997)](https://doi.org/10.1103/PhysRevB.56.12847)
+- [*Maximally localized Wannier functions for entangled energy bands*, Souza, Marzari and Vanderbilt, Phys. Rev. B **65**, 035109 (2001)](https://doi.org/10.1103/PhysRevB.65.035109)
+- [*Maximally localized Wannier functions: Theory and applications*, Marzari *et. al.*, Rev. Mod. Phys. **84**, 1419 (2012)](https://doi.org/10.1103/RevModPhys.84.1419)
+- [*Introduction to Maximally Localized Wannier Functions*, Ambrosetti and Silvestrelli, Reviews in Computational Chemistry, Ch. 6, pp. 327 (2016)](https://doi.org/10.1002/9781119148739.ch6)