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$\mathrm{LiHoF_{4}}$ is a magnetic material known for its Ising-typeanisotropy, making it a model system for studying quantum magnetism. However,the theoretical description of $\mathrm{LiHoF_{4}}$ using the quantum Isingmodel has shown discrepancies in its phase diagram, particularly in the regimedominated by thermal fluctuations. In this study, we investigate the role ofoff-diagonal dipolar terms in $\mathrm{LiHoF_{4}}$, previously neglected, indetermining its properties. We analytically derive the low-energy effectiveHamiltonian of $\mathrm{LiHoF_{4}}$, including the off-diagonal dipolar termsperturbatively, both in the absence and presence of a transverse field. Ourresults encompass the full $B_{x}-T$ phase diagram, confirming the significanceof the off-diagonal dipolar terms in reducing the zero-field criticaltemperature and determining the critical temperature's dependence on thetransverse field. We also highlight the sensitivity of this mechanism to thecrystal structure by comparing our calculations with the $\mathrm{Fe_{8}}$system.

LiHoF4 as a spin-half non-standard quantum Ising system