  The intrinsic anomalous Hall effect 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 the anomalous Hall conductivity. 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
set up 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 more than 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.
