Fourier series

Fourier series represent functions as linear combinations of sinusoids whose frequencies are integer multiples of a fundamental mode. In Wannier interpolation, the periodicity of an operator, $\hat{O}$, in a crystal admits a Fourier series expansion of the Bloch operator $\hat{O}(\bm{k}) = e^{-i\bm{k}\cdot\hat{\bm{r}}} \hat{O} e^{i\bm{k}\cdot\hat{\bm{r}}}$ as linear combinations of sinusoids at integer multiples of the real-space lattice vectors, forming a Bravais lattice. When exponentially localized Wannier functions exist, it suffices to use a modest number of lattice vectors in this expansion. The goal of this page of documentation is to describe the features, interface, and conventions of Fourier series evaluation as implemented by this library.

Lattice vectors

It is conventional to construct a basis for a reciprocal lattice $\{\bm{b}_j\}$ from a basis for a real-space lattice $\{\bm{a}_i\}$ such that

\[\bm{b}_j \cdot \bm{a}_i = 2\pi\delta_{ij}\]

Then we write the momentum space variable in the reciprocal lattice vector basis

\[\bm{k} = \sum_{j=1}^{d} k_j \bm{b}_j\]

Without loss of generality, the $k_j$ can be taken in the domain $[0,1]$. Additionally, any real space lattice vector can be written as an integer linear combination of the $\{\bm{a}_i\}$. We lastly mention that when computing Brillouin zone integrals, we keep track of this change of basis with the Jacobian determinant of the transformation,

\[\int_{\text{BZ}} d \bm{k} = |\det{B}| \int_{[0,1]^3} d^3 k.\]

Here, $B$ is a matrix whose columns are the $\{\bm{b}_j\}$.

Series coefficients

Wannier-interpolated Hamiltonians are small matrices obtained by projecting the ab-initio Hamiltonian onto a subspace of bands with energies near the Fermi level, a process called downfolding. These Hamiltonians can be expressed by a Fourier series as in the sum below

\[H(\bm{k}) = \sum_{\bm{R}} e^{i\bm{k}\cdot\bm{R}} H_{\bm{R}}\]

where the coefficients $H_{\bm{R}}$ are the matrix-valued Fourier coefficients. Truncating the sum over $\bm{R}$ at a modest number of modes can be done for Wannier Hamiltonians in the maximally-localized orbital basis, for which $H(\bm{k})$ is a smooth and periodic function and thus the truncation error of its Fourier series converges super-algebraically with respect to the number of modes.

Hamiltonian recipe

In model systems, a Bloch Hamiltonian can often be written down analytically. The recipe to write it as a Fourier series has two-steps

Identify the real and reciprocal Bravais lattices, $\{\bm{a}_i\}$ and $\{\bm{b}_j\}$, and rewrite all of the phase dependences of the Hamiltonian as $\bm{k}\cdot\bm{R}$ with $\bm{k}$ written in the basis of $\{\bm{b}_j\}$ and the lattice vector $\bm{R}$ written as an integer linear combination of $\{\bm{a}_i\}$.
Factor the Hamiltonian into a linear combination of normal modes indexed by the distinct $\bm{R}$ vectors. If the Hamiltonian is matrix-valued, this can be done one matrix element at a time.

For a non-trivial example, see the DOS interpolation for Graphene.

Interface

See FourierSeriesEvaluators.jl for the AbstractFourierSeries interface, which allows evaluation of the series with a function-like f(x) syntax. The Wannier90 interface will automatically construct these objects from the necessary output files.

Types

The concrete types listed below all implement the AbstractFourierSeries interface and should cover most use cases. They are organized by several abstract types to provide indicate physical properties of the Wannier-interpolated operator, such as a coordinate system or a gauge.

AutoBZ.AbstractWannierInterp — Type

AbstractWannierInterp{N,T,iip} <: AbstractFourierSeries{N,T,iip}

Abstract supertype for all Wannier-interpolated quantities in AutoBZ