Quantum fluctuations in two-dimensional superconductors

For granular metallic films or two-dimensional arrays of Josephson junctions, fluctuations in the number of Cooper pairs are conjugate to the superconducting phase and can lead to zero-point destruction of phase coherence at zero temperature. In two dimensions the resulting critical behavior is shown to be that of the $\lambda$ point of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ in three dimensions, with the charging energy playing the role of temperature. The nature of the quantum → classical crossover and its effect on the frequency dependence of precursor diamagnetism near the quantum critical point are discussed.