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 ${T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T}_{p}(x)$ 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 $\ensuremath{\sim}[\mathrm{cos}({k}_{x}a/\ensuremath{\pi})\ensuremath{-}\mathrm{cos}({k}_{y}a/\ensuremath{\pi}){]}^{2}$. The properties in this phase are compared to the properties of the so-called ``pseudogap phase'' of the copper oxides.