Specific heat and nonlinear susceptibility in spin glasses with random fields

Abstract

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 ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$. 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 $\mathrm{\ensuremath{\Delta}}$. 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 ${\ensuremath{\chi}}_{3}$, and phase diagrams for different disorder configurations. In the absence of RF, the ${\ensuremath{\chi}}_{3}$ 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 $\mathrm{\ensuremath{\Delta}}$. However, the ${T}_{f}$ is displaced to lower temperatures, enhancing considerably the ratio ${T}^{**}/{T}_{f}$. Furthermore, the divergence in ${\ensuremath{\chi}}_{3}$ is replaced by a rounded maximum at a temperature ${T}^{*}$, which becomes increasingly higher than ${T}_{f}$ as $\mathrm{\ensuremath{\Delta}}$ 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 ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ as the rounded maximum of ${\ensuremath{\chi}}_{3}$ behavior triggered by RF and the deviation of the conventional relationship between the ${T}_{f}$ and ${T}^{**}$.