Monte Carlo study of a generalized icosahedral model on the simple cubic lattice

We study the critical behavior of a generalized icosahedral model on the simple cubic lattice. The field variable of the icosahedral model might take one of twelve vectors of unit length, which are given by the normalized vertices of the icosahedron, as value. Similar to the Blume-Capel model, where in addition to $-1$ and $1$, as in the Ising model, the spin might take the value $0$, we add in the generalized model $(0,0,0)$ as allowed value. There is a parameter $D$ that controls the density of these voids. For a certain range of $D$, the model undergoes a second-order phase transition. On the critical line, $O(3)$ symmetry emerges. Furthermore, we demonstrate that within this range, similar to the Blume-Capel model on the simple cubic lattice, there is a value of $D$, where leading corrections to scaling vanish. We perform Monte Carlo simulations for lattices of a linear size up to $L=400$ by using a hybrid of local Metropolis and cluster updates. The motivation to study this particular model is mainly of technical nature. Less memory and CPU time are needed than for a model with $O(3)$ symmetry at the microscopic level. As the result of a finite-size scaling analysis we obtain $\nu=0.71164(10)$, $\eta=0.03784(5)$, and $\omega=0.759(2)$ for the critical exponents of the three-dimensional Heisenberg universality class. The estimate of the irrelevant renormalization group eigenvalue that is related with the breaking the $O(3)$ symmetry is $y_{ico}=-2.19(2)$.