We study magnetic properties of spin glass (SG) systems under a random field (RF), based on the suggestion that RFs can be induced by a weak transverse field in the compound ${\mathrm{LiHo}}_x{\mathrm{Y}}_{1-x}{\mathrm{F}}_4$. We consider a cluster spin model that allows long-range disordered interactions among clusters and short-range interactions inside the clusters, besides a local RF for each spin following a Gaussian distribution with standard deviation $\mathrm{\Delta }$. We adopt the one-step replica symmetry breaking approach to get an exactly solvable single-cluster problem. We discuss the behavior of order parameters, specific heat $C_m$, nonlinear susceptibility ${\chi }_3$, and phase diagrams for different disorder configurations. In the absence of RF, the ${\chi }_3$ exhibits a divergence at $T_f$, while the $C_m$ shows a broad maximum at a temperature $T^{**}$ around $30\%$ above $T_f$, as expected for conventional SG systems. The presence of RF changes this scenario. The $C_m$ still shows the maximum at $T^{**}$ that is weakly dependent on $\mathrm{\Delta }$. However, the $T_f$ is displaced to lower temperatures, enhancing considerably the ratio $T^{**}/T_f$. Furthermore, the divergence in ${\chi }_3$ is replaced by a rounded maximum at a temperature $T^{*}$, which becomes increasingly higher than $T_f$ as $\mathrm{\Delta }$ is enhanced. As a consequence, the paramagnetic phase is unfolded in three regions: (i) a conventional paramagnetism ($T>T^{**}$); (ii) a region with formation of short-range order with frozen spins ($T^{*}<T<T^{**}$); (iii) a region with slow growth of free-energy barriers slowing down the spin dynamics before the SG transition ($T_f<T<T^{*}$) suggesting an intermediate Griffiths phase before the SG state. Our results reproduce qualitatively some findings of ${\mathrm{LiHo}}_x{\mathrm{Y}}_{1-x}{\mathrm{F}}_4$ as the rounded maximum of ${\chi }_3$ behavior triggered by RF and the deviation of the conventional relationship between the $T_f$ and $T^{**}$.

• Received 17 April 2018
• Revised 21 December 2018

DOI:https://doi.org/10.1103/PhysRevB.99.014203