Long-range Coulomb interactions and the onset of superconductivity in the high-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="italic">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">c</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>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.

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