Model for Dipolar Glass and Relaxor Ferroelectric Behavior

TL;DR

This study uses Monte Carlo simulations of a 12-state discretized Heisenberg model with random fields to explore phase transitions and ferroelectric order in dipolar glass and relaxor ferroelectric materials.

Contribution

It introduces a detailed simulation of a 12-state model with random fields, revealing phase transition behaviors and the influence of cubic anisotropy on ferroelectric order.

Findings

Identified phase transitions at specific temperatures for various random field strengths.

Observed ferroelectric order below transition temperatures, aligned along [111] directions.

Different critical behaviors suggest the presence of cubic and isotropic fixed points.

Abstract

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 $T_c$.