Josephson-Junction Arrays in Transverse Magnetic Fields

A class of uniformly frustrated $\mathrm{xy}$ models which describe the behavior of a Josephson-junction array in a transverse magnetic field is considered. The frustration $f$ is the fraction of flux quantum of applied field per unit cell of the lattice. The ground-state energy $E_0$ and critical current are computed for several rational $f$. The behavior for arbitrary $f$ is discussed, and it is concluded that the resistive transition temperature for $f=\frac{p}{q}$ is bounded by $k_{\mathrm{B}}{T}_c\left(\frac{p}{q}\right)\lesssim \frac{\pi \left|E_0\mathrm{}\right(f\left)\right|}{2q}$. The resistance as a function of temperature and field is determined by the sequence of $T_c$'s thus produced.