2022/06/14 by Fisch, Ronald
#Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.2206.06964
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 hR. The 12 states correspond to the [110] directions of a cube. Simple cubic lattices of size 128 × 128 × 128 with periodic boundary conditions were used, and 32 samples were studied for each value of hR. The model has the standard nonrandom two-spin exchange term with coupling energy J and a field which adds an energy hR to two of the 12 spin states, chosen randomly and independently at each site. We provide results for the cases hR / J = -2.5, -2.0, -1.5, 3.0 and 4.0. For all these cases except hR / J = -2.5, we see an apparently sharp phase transition at a temperature Tc where the specific heat and the longitudinal susceptibility are peaked. At Tc, 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 hR /J = -1.5 and 3.0 than it is for hR / 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 Tc, 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 hR / J = -2.5, we do not see clear evidence of a single well-defined Tc.