The two-dimensional ($2d$) fully frustrated Planar Rotator model on a square lattice has been the subject of a long controversy due to the simultaneous $Z_2$ and $O(2)$ symmetry existing in the model. The $O(2)$ symmetry being responsible for the Berezinskii - Kosterlitz - Thouless transition ($BKT$) while the $Z_2$ drives an Ising-like transition. There are arguments supporting two possible scenarios, one advocating that the loss of $Ising$ and $BKT$ order take place at the same temperature $T_{t}$ and the other that the $Z_2$ transition occurs at a higher temperature than the $BKT$ one. In the first case an immediate consequence is that this model is in a new universality class. Most of the studies take hand of some order parameter like the stiffness, Binders cumulant or magnetization to obtain the transition temperature. Considering that the transition temperatures are obtained, in general, as an average over the estimates taken about several of those quantities, it is difficult to decide if they are describing the same or slightly separate transitions. In this paper we describe an iterative method based on the knowledge of the complex zeros of the energy probability distribution to study the critical behavior of the system. The method is general with advantages over most conventional techniques since it does not need to identify any order parameter emph{a priori}. The critical temperature and exponents can be obtained with good precision. We apply the method to study the Fully Frustrated Planar Rotator ($PR$) and the Anisotropic Heisenberg ($XY$) models in two dimensions. We show that both models are in a new universality class with $T_{PR}=0.45286(32)$ and $T_{XY}=0.36916(16)$ and the transition exponent $ u=0.824(30)$ ($frac{1}{ u}=1.22(4)$).
تحميل البحث