[27] Efficiency of the microcanonical over-relaxation algorithm for vector spins analyzing first and second order transitions

We simulate vectorial spin systems solely with the microcanonical over--relaxation algorithm where the temperature is calculated by a formula of Nurdin and Schotte. We show that this procedure is the most efficient local algorithm besides the nonlocal cluster algorithm not only for first order transitions but also for second order ones. A comparison is made with the Metropolis, heat bath, multicanonical and the Creutz's demon algorithms. We study, using these algorithms, the frustrated $J_1$--$J_2$ model on a cubic lattice for $XY$, Heisenberg and $O(4)$ spins. These models have a breakdown of symmetry $Z_3 \otimes SO(N)/SO(N-1)$ for the number $N = 2,\,3,\,4\,$ of spin components leading to transitions of first order. We show that they are strongly first order. Then, to test the over--relaxation update for second order transitions, we study a ferromagnet on a cubic lattice and a frustrated antiferromagnet on a stacked triangular lattice. We finally point out the advantages and the flaws of the over--relaxation procedure.