Doping-driven transition to a time-reversal breaking state in the phase diagram of the cuprates

Motivated by recent tunneling and Andreev-reflection experiments, we study the conditions for a quantum transition within the superconducting phase of the cuprates, in which a bulk imaginary (time-reversal breaking) id(xy) component appears in addition to the d(x)(2)-y(2) order parameter. We examine in detail the role of some important physical features of the cuprates. In particular we show that a closed Fermi surface, a bilayer splitting, an orthorhombic distortion, and the proximity to a quantum critical point around optimal doping favor the appearance of the imaginary component. These findings could explain why the mixed d(x)(2)-y(2)+id(xy) order parameter is observed in Y1-yCayBa2Cu3O7-x and La2-xSrxCuO4, and suggest that it could appear also in Bi2Sr2CaCu2O8. We also predict that, in all cuprates, the mixed state should be stable only in a limited doping region all contained beneath the d(x)(2)-y(2) dome. The behavior of the specific heat at the secondary transition is discussed.