Superconducting Coherence and the Helicity Modulus in Vortex Line Models

We show how commonly used models for vortex lines in three dimensional superconductors can be modified to include k=0 excitations. We construct a formula for the k=0 helicity modulus in terms of fluctuations in the projected area of vortex loops. This gives a convenient criterion for the presence of superconducting coherence. We also present Monte Carlo simulations of a continuum vortex line model for the melting of the Abrikosov vortex lattice in pure YBCO.