Superconducting, metallic, and insulating phases in a model of ${\mathrm{CuO}}_2$ layers

We examine a three-band model of the ${\mathrm{CuO}}_2$ layers in cuprate superconductors in a systematic large-N expansion obtained by generalizing the SU(2) spins of the model to symplectic Sp(N) symmetry. We find superconducting and metallic ground states connected by a zero-temperature, superconductor-metal transition (SMT) at finite doping. Qualitative features of the temperature-doping phase diagram are consistent with experiments. The critical properties of the SMT are shown to be described by a continuum model of spin-1/2, charge-e fermions, and a charge-(-e) scalar. A renormalization-group analysis of this model finds either a second-order SMT or runaway renormalization-group flows, usually interpreted as a fluctuation-induced first-order transition. The flux quantum of this model remains at hc/2e, although hc/e vortices should become stable near the SMT unless the SMT is strongly first order.