Long-range Coulomb interactions and the onset of superconductivity in the high-${\mathit{T}}_{\mathit{c}}$ materials

The Ginzburg-Landau functional for a superconductor is extended to include a quantum-fluctuation term arising from imperfect screening of the long-range Coulomb interaction. At low temperatures the resulting quantum x-y model shows a second-order phase transition between a superconducting state and an insulating state as a function of the ratio of the phase stiffness to the Coulomb energy measured on the scale of the mean pair spacing. By relating the functional formulation to a BCS-type model of high-temperature superconductivity in the strongly correlated regime, we show that the phase stiffness is proportional to doping away from the 1/2-full Mott insulating state. We discuss application of the model as a mechanism for the onset of superconductivity of the ${\mathrm{CuO}}_2$-based high-${\mathit{T}}_{\mathit{c}}$ materials above a critical doping level. Transport and optical properties of materials with reduced transition temperature are calculated.