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

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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) ${\mathrm{id}}_{\mathrm{xy}}$ component appears in addition to the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ 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 ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}{+id}_{\mathrm{xy}}$ order parameter is observed in ${\mathrm{Y}}_{1\ensuremath{-}y}{\mathrm{Ca}}_{y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}x}$ and ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4},$ and suggest that it could appear also in ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8}.$ We also predict that, in all cuprates, the mixed state should be stable only in a limited doping region all contained beneath the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ dome. The behavior of the specific heat at the secondary transition is discussed.