Model for Dipolar Glass and Relaxor Ferroelectric Behavior

Heat bath Monte Carlo simulations have been used to study a 12-state discretized Heisenberg model with a type of random field, for several values of the randomness coupling parameter $h_R$. The 12 states correspond to the [110] directions of a cube. Simple cubic lattices of size $128 \times 128 \times 128$ with periodic boundary conditions were used, and 32 samples were studied for each value of $h_R$. The model has the standard nonrandom two-spin exchange term with coupling energy $J$ and a field which adds an energy $h_R$ to two of the 12 spin states, chosen randomly and independently at each site. We provide results for the cases $h_R / J =$ -2.5, -2.0, -1.5, 3.0 and 4.0. For all these cases except $h_R / J =$ -2.5, we see an apparently sharp phase transition at a temperature $T_c$ where the specific heat and the longitudinal susceptibility are peaked. At $T_c$, the behavior of the peak in the structure factor, $S ({\bf k} )$, at small $|{\bf k}|$ is a straight line on a log-log plot. However, the value of the slope of this line is different for $h_R /J =$ -1.5 and 3.0 than it is for $h_R / J =$ -2.0 and 4.0. We believe that the first two cases are showing the behavior of a cubic fixed point in a weak random field, and the behavior of the second two cases are showing the behavior of an isotropic fixed point when the Imry-Ma length is smaller than the sample size. Below $T_c$, these $L = 128$ samples show ferroelectric order, and this order rapidly becomes oriented along one of the eight [111] directions as $T$ is reduced. This rotation of the ordering direction is caused by the cubic anisotropy. For $h_R / J =$ -2.5, we do not see clear evidence of a single well-defined $T_c$.