We theoretically investigate the influence of off-diagonal dipolar interactions in the mixed-anisotropy magnet ${\mathrm{LiHo}}_x{\mathrm{Er}}_{1-x}{\mathrm{F}}_4$. Motivated by experimental observations showing an unexpectedly rapid suppression of the ferromagnetic transition temperature $T_c$ upon substitution of Ho by Er ions, we use Monte Carlo simulations incorporating off-diagonal dipolar terms to elucidate the underlying physical mechanism. Our results reveal that, unlike in the diluted magnet ${\mathrm{LiHo}}_x{\mathrm{Y}}_{1-x}{\mathrm{F}}_4$, where dilution weakens the effect of these interactions, substitution by planar Er ions amplifies it. This leads to a pronounced reduction of $T_c$, closely matching experimental data, thereby providing a natural explanation for discrepancies with mean-field predictions. The findings underscore the essential role of off-diagonal dipolar interactions in determining the magnetic properties of mixed-anisotropy dipolar systems.

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• Dipolar interaction
• Magnetic anisotropy
• Magnetism
• Ising model
• Monte Carlo methods
• XY model