Time Reversal Breaking Superconducting State in the Phase Diagram of the Cuprates

We review and extend a previous study on the symmetry of the superconducting state, stimulated by recent tunneling and Andreev reflection measurements giving robust evidences for the existence of a $d_{x^2-y^2}+id_{xy}$ order parameter in the overdoped regime of two different cuprates. Looking for a possible second-order phase transition from a standard $d_{x^2-y^2}$ to a mixed and time reversal breaking state, we confirm the results of our previous analysis on La$_{2-x}$Sr$_{x}$CuO$_{4}$. In the case of Y$_{1-y}$Ca$_y$Ba$_2$Cu$_3$O$_{7-x}$ as well, among all the allowed symmetries, the $d_{x^2-y^2}+id_{xy}$ is the most favored one and the unconventional state is likely to occur in a small dome of the phase diagram located in the optimal-overdoped region and at very low temperatures.