### abstract

- The random-field Ising model (RFIM) is central to the study of disordered systems. Yet, for a long time it eluded realization in ferromagnetic systems because of the difficulty to produce locally random magnetic fields. Recently, it was shown that in anisotropic dipolar magnetic insulators, the archetype of which is the ${\text{LiHo}}_{x}{\text{Y}}_{1\ensuremath{-}x}{\text{F}}_{4}$ system, the RFIM can be realized in both ferromagnetic and spin-glass phases. The interplay between an applied transverse field and the off-diagonal terms of the dipolar interaction produce effective longitudinal fields, which are random in sign and magnitude as a result of spatial dilution. In this paper, we use exact numerical diagonalization of the full Hamiltonian of Ho pairs in ${\text{LiHo}}_{x}{\text{Y}}_{1\ensuremath{-}x}{\text{F}}_{4}$ to calculate the effective longitudinal field beyond the perturbative regime. In particular, we find that nearby spins can experience an effective field larger than the intrinsic dipolar broadening (of quantum states in zero field) which can therefore be evidenced in experiments. We then calculate the magnetization and susceptibility under several experimental protocols, and show how these protocols can produce direct measurement of the effective longitudinal field.