of the grid, $A_{i\mu}$ is proportional to the magnetic vector potential integrated across the bond, and the argument of the cosine is thus the gauge invariant phase angle difference across the bond.

The $A_{\rm i\mu}$ are determined such that the circulation of $A_{i\mu}$ around any plaquette $\alpha$ of the grid is fixed and equal to $2\pi f_\alpha$ with $f_\alpha$ equal to the fraction of applied magnetic flux quanta through that plaquette. To model a uniform applied magnetic field in the $\hat z$ direction, we use a uniform value $f_\alpha = f$ for plaquettes oriented with normal in the $\hat z$ direction, and $f_\alpha =0$ otherwise. The presence of the applied magnetic field induces vortex lines into the phase angles $\theta({\bf r}_i)$ such that the average vortex line density in direction $\hat \mu$ is $n_\mu=f\delta_{\mu z}$. Holding the $A_{i\mu}$ fixed corresponds to the approximation of an infinite magnetic penetration length $\lambda$, which one may expect to be reasonable in the limit that $\lambda$ is much larger than the average inter-vortex spacing. Experiments on very pure YBCO samples[@Liang] indicate that many features of behavior, up to surprisingly large magnetic fields, are well described by this large $\lambda\to\infty$ XY model limit. We return 