On vortices and phase coherence in high-T_c superconductors

An array of Cooper paired planes will not have long-range phase coherence at any finite temperature owing to an infrared divergence of phase fluctuations when the coherence length perpendicular to the plane is small enough to prevent leakage of the superconducting order parameter within the planes. The phase correlations decay in a sufficiently slow manner to provide enough local phase coherence to make possible the nucleation of vortices. The planes then acquire Kosterlitz-Thouless topological order with its intrinsic rigidity and concomitant superfluidity. We conclude that the high-temperature superconducting cuprates are topologically ordered superconductors rather than phase-ordered superconductors since the large insulating layer between the copper-oxygen planes prevents effective leakage of the superfluid order parameter. For low enough superfluid densities, as in the underdoped cuprates studied by Uemura, the transition temperature T_c will be proportional to the superfluid density corresponding to vortex-antivortex unbinding, and not to the disappearance of the Cooper pairing amplitude. Above T_c, but below the Bardeen-Cooper-Schrieffer pairing temperature T_p, we shall have a dephased Cooper pair fluid that is a vortexantivortex liquid. Since the superconductivity is effectively two dimensional, there can be a large difference between T_p and T_c, as observed in the underdoped cuprates. The ac and dc conductivities measured by Corson et al. in this region are those corresponding to flux flow. Furthermore there will be vortices over a large temperature region above T_c which will lead to a Nernst vortex-like response and there will be a measurable depairing field H_c2 above T_c as evidenced by recent experiments by Wang et al.