The 2D J_1-J_2 XY and XY-Ising Models

We consider the 2D $J_1-J_2$ classical XY model on a square lattice. In the frustrated phase corresponding to $J_2>J_1/2$, an Ising order parameter emerges by an ``order due to disorder'' effect. This leads to a discrete symmetry plus the O(2) global one. We formulate the problem in a Coulomb gas language and show by a renormalization group analysis that only two phases are still possible : a locked phase at low temperature and a disordered one at high temperature. The transition is characterized by the loss of Ising and XY order at the same point. This analysis suggests that the 2D $J_1-J_2$ XY model is in the same universality class than XY-Ising models.