Fourier transforms
The Fourier-space frequencies for Wannier interpolation, also called $\\mathbf{R}$-vectors.
The Wannier interpolation is a Fourier transform of a (theoretically continuous periodic function (Bloch wavefunction) into a Fourier series. However, since the wavefunctions are discretized in $\\mathbf{k}$-space on a grid, the Fourier series of a coutinuous functionis turns into a discrete-time Fourier transform (DTFT). For the usual DTFT, the signal is transformed from time domain to frequency domain. Here, the time domain is the $\\mathbf{k}$-space, while the frequency domain is the $\\mathbf{R}$-space.
To improve interpolation accuracy, the frequencies (often they are called the $\\mathbf{R}$-vectors, since they live in 3D space) are not chosen inside a parallelepiped, but inside a Wigner-Seitz cell to make sure most of the large-magnitude interactions (e.g., Hamiltonian $H(\\mathbf{R})$) are included. Therefore, we have some special algorithms to generate the $\\mathbf{R}$-space domain.
Th function [generate_Rspace] can generate two kinds of R-space domains, by providing an argument of type WannierInterpolationAlgorithm:
WSInterpolation: Wigner-Seitz interpolation, returns aWignerSeitzRspaceMDRSInterpolation: minimal-distance replica selection interpolation, returns aMDRSRspace
However, the details of how to place the R-vectors and setting their degeneracies are irrelevant to the Fourier / inverse Fourier transforms, therefore, we provide a function simplify that can convert each kind of RspaceDomain to a BareRspace, enabling faster Fourier transforms.
Depending on the type of RspaceDomain, there are three kinds of Fourier transform:
BareRspace: simple Fourier summath O_{mn}(\\mathbf{R}) = \\frac{1}{N_{\\mathbf{k}}} \\sum_{\\mathbf{k}} \\exp(-i {\\mathbf{k}} \\mathbf{R}) O_{mn}(\\mathbf{k}),WignerSeitzRspace: Wigner-Seitz interpolation, the forward Fourier transform is the same asBareRspace,MDRSRspace: minimal-distance replica selection interpolationmath O_{mn}(\\widetilde{ \\mathbf{R} }) = \\sum_{ \\mathbf{R} } \\frac{1}{\\mathcal{N}_{\\mathbf{R}} \\mathcal{N}_{mn \\mathbf{R}}} O_{mn}(\\mathbf{R}) \\sum_{i=1}^{\\mathcal{N}_{mn \\mathbf{R}}} \\delta_{ \\widetilde{\\mathbf{R}}, \\mathbf{R} + \\mathbf{T}_{mn \\mathbf{R}}^{(i)} },
where
- $N_{\\mathbf{k}}$: the total number of kpoints
- $\\mathcal{N}_{\\mathbf{R}}$: the degeneracy of R-vectors
- $\\mathcal{N}_{mn \\mathbf{R}}$ is the degeneracy of $\\mathbf{T}_{mn \\mathbf{R}}$ vectors
invfourier
WS
\[O_{mn}(\\mathbf{k}) = \\sum_{\\mathbf{R}} \\frac{1}{\\mathcal{N}_{\\mathbf{R}}} \\exp(i \\mathbf{k} \\mathbf{R}) O_{mn}(\\mathbf{R}),\]
where $\\mathcal{N}_{\\mathbf{R}}$ is the degeneracy of R vectors (not the total number of R vectors).
MDRS
\[X_{mn}(\\mathbf{k}) = \\sum_{\\mathbf{R}} \\frac{1}{ \\mathcal{N}_{mn \\mathbf{R}} } X_{mn}(\\mathbf{R}) \\sum_{j=1}^{\\mathcal{N}_{ mn \\mathbf{R} }} \\exp\left( i \\mathbf{k} \cdot \left( \\mathbf{R} + \\mathbf{T}_{ mn \\mathbf{R} }^{(j)} \right) \right)\]
where $\\mathcal{N}_{\\mathbf{R}}$ is the degeneracy of R vectors (not the total number of R vectors), and $\\mathcal{N}_{ mn \\mathbf{R} }$ is the degeneracy of $\\mathbf{T}_{ mn \\mathbf{R} }$ vectors.
MDRSv2
\[X_{mn}(\\mathbf{k}) = \\sum_{ \\widetilde{\\mathbf{R}} } \\exp\left( i \\mathbf{k} \\widetilde{ \\mathbf{R} } \right) \\widetilde{X}_{mn}(\\widetilde{\\mathbf{R}})\]