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 T0ex0ex=0ex0exTₚ(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 ~[cos(kₓa/π)-cos(kya/π)]². The properties in this phase are compared to the properties of the so-called ``pseudogap phase'' of the copper oxides.