Monte Carlo simulation on the first-order melting transition of high-$T_c$ superconductors in $\mathbf{B}\Vert \mathbf{c}^$

The first-order melting transition of the vortex-line lattice of high-$T_c$ superconductors in $\mathbf{B}\mathbf{\Vert }\mathbf{ĉ}$ is investigated by Monte Carlo simulation based on the three-dimensional uniformly frustrated, anisotropic $\mathrm{XY}$ model. A $\delta$-function peak in the specific heat is observed at the melting point $T_m$ associated with a latent heat, where the triangular vortex-line lattice melts. A jump in the helicity modulus along the $c$ axis is observed at the melting point from zero to $16{\pi }^3{\lambda }_{\mathrm{ab}}^2{(T}_m\left){{\rm Y}}_c{(T}_m\right){\Gamma }^2/\left(d{\phi }_0^2\right)\simeq 0.6$ with $\Gamma$ the anisotropy constant in the present model. Therefore, the system is superconducting only below $T_m$ and along the $c$ axis. There is, however, short-range correlation between vortices both along the $c$ axis and in the $\mathrm{ab}$ plane, and therefore the system behaves as a liquid of vortex lines, in the vicinity above the melting point for the anisotropy constants studied. The simulation gives estimates on the jumps of entropy and magnetic induction on the melting in good agreement with the experimental observation on a ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ single crystal. Although the agreement is not as good as with the ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ sample, the simulation results are comparable with the experimental observation on a ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$ sample. The simulated melting line in the $B-T$ diagram is fitted very well by $B_m=0.132\mathrm{}\times \{d{\phi }_0^2/\left[16{\pi }^3{\lambda }_{\mathrm{ab}}^2{(T}_m{)k}_B{T}_m\right]{\}}^2,$ which coincides with the melting theory based on a continuum description of the vortex-line lattice combined with the Lindemann criterion taking $c_L=0.18\mathrm{}$.