commensurability effects (for example a square rather than a triangular ground state vortex lattice), one may still hope that many features of our system with respect to vortex lattice melting and the vortex glass transition will remain qualitatively similar to results on real physical materials.

To probe the behavior of the system we consider the average energy per site, which is the thermodynamic conjugate variable to the temperature $T$, $$\begin{aligned}
E&\equiv& {1\over L^2L_z}{\partial (\beta {\cal F})\over \partial \beta}\\ \nonumber
&=&-{1\over L^2L_z}\sum_{i,\mu}J_{i\mu}\langle \cos(\theta({\bf r}_i)-\theta({\bf r}_i+\hat\mu)-A_{i\mu})\rangle \enspace,\end{aligned}$$ where $\beta\equiv1/T$ and ${\cal F}$ is the total free energy. We can similarly define a variable $Q$ which is the thermodynamic conjugate to the disorder strength $p$, $$\begin{aligned}
Q&\equiv&{1\over L^2L_z}{\partial {\cal F}\over\partial p}\\ \nonumber
&=&-{1\over L^2L_z}\sum_{i,\mu=x,y}J_\perp\epsilon_{i\mu}\langle\cos(\theta({\bf r}_i)-\theta({\bf r}_i+\hat\mu)-A_{i\mu})\rangle \enspace.\end{aligned}$$ The vorticity in the system is determined by considering the circulation of the gauge invariant phase difference around each plaquette. For a plaquette $\alpha$ at position ${\bf r}_\alpha$ with normal in direction $\mu$, the vorticity $n_\mu({\bf r}_\alpha)$ piercing the plaquette is determined by, $$\sum_\alpha \left[\theta({\bf r}_i)-\theta({\bf r}_i+\hat\mu)-A_{i\mu}\right]_{-\pi}^{+\pi} = 2\pi[n_\mu({\bf r}_\alpha)-f\delta_{\mu z}]\enspace,$$ where the sum goes counterclockwise around the bonds of the plaquette, and the bracket on the left hand side indicates that the gauge invariant phase 