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

A systematic solution of a model for copper oxides reveals a line of transi tions T = T-p(x.) for x, the doping away from half filling less than a crit ical value, to a phase with broken time reversal and rotational symmetry. T he single-particle spectrum in this phase is calculated to have a gap simil ar to[cos(k(x)a/pi) - cos(k(y)a/pi)](2). The properties in this phase are c ompared to the properties of the so-called "pseudogap phase" of the copper oxides.