  The properties of LiHoF$_4$ are believed to be well described by a long-range
dipolar Ising model. We go beyond mean-field theory and calculate the phase
diagram of the Ising model in a transverse field using a quantum Monte Carlo
method. The relevant Ising degrees of freedom are obtained using a
non-perturbative projection onto the low-lying crystal field eigenstates. We
explicitly take the domain structure into account, and the strength of the
near-neighbor exchange interaction is obtained as a fitting parameter. The
on-site hyperfine interaction is approximately taken into account through a
renormalization of the transverse applied magnetic field. Finally, we propose a
spectroscopy experiment to precisely measure the most important parameter
controlling the location of the phase boundary.
