  Using a realistic tight-binding Hamiltonian based on maximally localized
Wannier functions, we investigate the two-ion magnetocrystalline anisotropy
energy (MAE) in $L1_0$ transition metal compounds. MAE contributions from
throughout the Brillouin zone are obtained using magnetic force theorem
calculations with and without perturbation theory. The results from either
method agree with one another, and both reflect features of the Fermi surface.
The intra-sublattice and inter-sublattice contributions to MAE are evaluated
using a Green's function method. We find that the sign of the inter-sublattice
contribution varies among compounds, and that its amplitude may be significant,
suggesting MAE can not be resolved accurately in a single-ion manner. The
results are further validated by scaling spin-orbit-coupling strength in
density functional theory. Overall, this realistic tight-binding method provide
an effective approach to evaluate and analyze MAE while retaining the accuracy
of corresponding first-principles methods.
