Effective spin-12description of transverse-field-induced random fields in dipolar spin glasses with strong single-ion anisotropy

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We present analytical and numerical evidence for the validity of an effective ${S}_{\mathrm{eff}}=\frac{1}{2}$ approach to the description of random field generation in $S\ensuremath{\geqslant}1$, and especially for $S=1$, dipolar spin-glass models with strong uniaxial Ising anisotropy and subject to weak external magnetic field ${B}_{x}$ transverse to the Ising direction. Explicitly ${B}_{x}$-dependent random fields are shown to naturally emerge in the effective low-energy description of a microscopic $S=1$ toy model. We discuss our results in relation to recent theoretical studies pertaining to the topic of ${B}_{x}$-induced random fields in $\mathrm{Li}{\mathrm{Ho}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ magnetic materials with the ${\mathrm{Ho}}^{3+}$ Ising moments subject to a transverse field. We show that the ${S}_{\mathrm{eff}}=\frac{1}{2}$ approach is able to capture both the qualitative and quantitative aspects of the physics at small ${B}_{x}$, giving results that agree with those obtained using conventional second-order perturbation theory.