Specific heat and nonlinear susceptibility in spin glasses with random fields

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 LiHoxY1-xF4. 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 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 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 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 LiHoxY1-xF4 as the rounded maximum of chi(3) behavior triggered by RF and the deviation of the conventional relationship between the T-f and T**.