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We present a general framework for deriving effective spin Hamiltonians ofcorrelated magnetic systems based on a combination of relativistic ab initiodensity functional theory calculations (DFT), exact diagonalization of ageneralized Hubbard Hamiltonian on finite clusters and spin projections ontothe low-energy subspace. A key motivation is to determine anisotropic bilinearexchange couplings in materials of interest. As an example, we apply thismethod to the pyrochlore Lu$_2$V$_2$O$_7$ where the vanadium ions form alattice of corner-sharing spin-1/2 tetrahedra. In this compound, anisotropicDzyaloshinskii-Moriya interactions (DMI) play an essential role in inducing amagnon Hall effect. We obtain quantitative estimates of the nearest-neighborHeisenberg exchange, the DMI, and the symmetric part of the anisotropicexchange tensor. Finally, we compare our results with experimental ones on theLu$_2$V$_2$O$_7$ compound.

Ab initio determination of spin Hamiltonians with anisotropic exchange interactions: the case of the pyrochlore ferromagnet Lu$_2$V$_2$O$_7$