Quantum melting of a two-dimensional vortex lattice at zero temperature

We consider the quantum melting of a two-dimensional flux lattice at temperature $T=0\mathrm{}$ in the "superclean limit." In this regime, we find that vortex motion is dominated by the Magnus force. A Lindemann criterion predicts melting when $\frac{n_v}{n_p}>~\beta$, where $n_v$ and $n_p$ are the areal number densities of vortex pancakes and Cooper pairs, and $\beta \approx 0.1$. A second criterion is derived by using Wigner-crystal and Laughlin wave functions for the solid and liquid phases respectively, and setting the two energies equal. This gives a melting value similar to the Lindemann result. We discuss the numerical value of the $T=0\mathrm{}$ melting field for thin layers of a low-$T_c$ superconductor, such as $a$-MoGe, and single layers of high-$T_c$ materials.