Lower critical dimensions for superconducting long-range order in type-II superconductors

It is shown that phase fluctuations destroy off-diagonal long-range order (phase coherence) in the Meissner phase below two dimensions, below four dimensions in the conventional Abrikosov flux-lattice phase, and below three dimensions in the limit of infinite Ginzburg ratio κ. The destruction of long-range order in the flux lattice is due to phase changes induced by shear motions of the flux lines. The phase coherence decays over a long length scale (of order millimeters) in three-dimensional systems