Pseudogap phase and the quantum-critical point in copper-oxide metals

A systematic solution of a 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. The single-particle spectrum in this phase is calculated to have a gap similar to{[}cos(k(x)a/pi) - cos(k(y)a/pi)](2). The properties in this phase are compared to the properties of the so-called ``pseudogap phase{''} of the copper oxides.