Comment on "Longitudinal Superconductivity in Vortex-Line Phases: A Monte Carlo Study"

Comment on " Longitudinal Superconductivity in Vortex-Line Phases: A Monte Carlo Study " In a recent Letter [1], Carneiro reports on new simulations of vortex-line fluctuations in a London type II su-perconductor with a finite magnetic penetration length l. He concludes that superconductivity parallel to the applied magnetic field H (" longitudinal superconductiv-ity ") vanishes at the same temperature as vortex-line lattice melting. This conclusion contradicts similar finite l simulations we reported on earlier [2], where we found evidence that (for L z , f 2 0 ͞2p 2 T m) longitudinal super-conductivity vanished at a T c distinctly above the melting T m. One main difference between the two simulations is that Carneiro uses an ensemble at fixed applied H, while we used an ensemble at fixed average internal. Carneiro's simulations are for a higher vortex-line density than our own; this alone might lead to the apparent merger of T c and T m , as we have observed in earlier l ! ` simulations [3]. However, we believe there is a more fundamental error in Carneiro's analysis. Carneiro uses as his criterion for the presence of longitudinal superconductivity the vanishing of the transverse magnetic susceptibility m 2 ϵ ͑≠B 2 ͞≠H 2 ͒ j H෇Hˆx 3. m 2 is proportional to the transverse fluctuation in vortex-line density n ෇ B͞f 0. Carneiro states that " If the system behaves like a superconductor for currents parallel to the external field.. . , a field perpendicular to it is shielded by the Meissner effect and m 2 vanishes. " This, however, is incorrect. For a periodic system with no true surface, such as Carneiro's model, adding a uniform transverse magnetic field does not imply the existence of current flowing parallel to the original field. It represents merely a tilting of the original field, and thermodynamic arguments [4] show that m 2 is related to the tilt modulus at zero wave vector , m 2 ෇ B 2 ͞4pc 44 ͑0͒. For a uniform superconductor (no pinning), c 44 ͑0͒ is finite in both vortex-line lattice and vortex-line liquid. Hence m 2 should nowhere vanish, and so is not a measure of superconductivity at all. The rapid rise in m 2 that Carneiro observes at T m is, we believe, a consequence of the fact that, for such high vortex-line densities , vortex-lattice melting coincides with a depinning of the vortex …