Persistent current in a 2D Josephson junction array wrapped around a   cylinder
  We study persistent currents in a Josephson junction array wrapped around a cylinder. The $T=0$ quantum statistical mechanics of the array is equivalent to the statistical mechanics of a classical $xy$ spin system in 2+1 dimensions at the effective temperature $T^{*}=\sqrt{2JU}$, with $J$ being the Josephson energy of the junction and $U$ being the charging energy of the superconducting island. It is investigated analytically and numerically on lattices containing over one million sites. For weak disorder and $T^{*}\ll J$ the dependence of the persistent current on disorder and $T^{*}$ computed numerically agrees quantitatively with the analytical result derived within the spin-wave approximation. The high-$T^{*}$ and/or strong-disorder behavior is dominated by instantons corresponding to the vortex loops in 2+1 dimensions. The current becomes destroyed completely at the quantum phase transition into the Cooper-pair insulating phase. 
D. A. Garanin and E. M. Chudnovsky
cond-mat.supr-con