Quantum corrections to the Ising interactions in LiYHoF4
  We systematically derive an effective two-state Hamiltonian for the dipolar Ising magnet LiYHoF4, including quantum corrections which arise from the transverse dipolar and hyperfine interactions. These corrections are derived using a generalised Schrieffer-Wolff transformation to leading order in the small parameters given by the ratio of the interaction energies to the energy of the first excited electronic state of the Holmium ions. The resulting low-energy Hamiltonian involves two-level systems, corresponding to the low-lying electronic states of the Holmiums, which are coupled to one another and to the Holmium nuclei. It differs from that obtained by treating the electronic states of the Holmium as a spin-1/2 with an anisotropic g-factor. It includes effective on-site transverse fields, and both pairwise and three-body interactions among the dipoles and nuclei. We explain the origins of the terms, and give numerical values for their strengths. 
A. Chin, P. R. Eastham