Ab initio calculation of the anomalous Hall conductivity by Wannier interpolation

The intrinsic anomalous Hall conductivity in ferromagnets depends on subtle spin-orbit-induced effects in the electronic structure, and recent ab initio studies found that it was necessary to sample the Brillouin zone at millions of $k$-points to converge the calculation. We present an efficient first-principles approach for computing this quantity. We start out by performing a conventional electronic-structure calculation including spin-orbit coupling on a uniform and relatively coarse $k$-point mesh. From the resulting Bloch states, maximally localized Wannier functions are constructed which reproduce the ab initio states up to the Fermi level. The Hamiltonian and position-operator matrix elements, needed to represent the energy bands and Berry curvatures, are then set up between the Wannier orbitals. This completes the first stage of the calculation, whereby the low-energy ab initio problem is transformed into an effective tight-binding form. The second stage only involves Fourier transforms and unitary transformations of the small matrices setup in the first stage. With these inexpensive operations, the quantities of interest are interpolated onto a dense $k$-point mesh and used to evaluate the anomalous Hall conductivity as a Brillouin zone integral. The present scheme, which also avoids the cumbersome summation over all unoccupied states in the Kubo formula, is applied to bcc Fe, giving excellent agreement with conventional, less efficient first-principles calculations. Remarkably, we find that about 99% of the effect can be recovered by keeping a set of terms depending only on the Hamiltonian matrix elements, not on matrix elements of the position operator.