Pseudo-Gap Phase, the Quantum-Critical Point and superconductivity in Copper-Oxide Metals

A systematic solution of a general model for Copper-Oxides reveals a line of transitions $T = T_p (x)$ for $x$, the doping away from half-filling less than a critical value, to a phase with broken time reversal and rotational symmetry but with translational symmetry preserved. The single-particle spectrum in this phase is calculated to have a gap of $d_{x^2 - y^2}$ symmetry. The properties in this phase are compared to the properties of the so called ''pseudo-gap phase'' in the Copper-Oxides. The fluctuations towards this phase promote superconductivity which is either of the $d_{x^2-y^2}$ or the generalised s-symmetry depending on whether the projected density of states at the chemical potential is larger in the $d_{x^2-y^2}$ channel, as in the hole-doped copper-oxides, or in the s-wave channel.