Phase coherence and the boson analogy of vortex liquids

The statistical mechanics of the flux-line lattice in extreme type-II superconductors is studied within the framework of the uniformly frustrated anisotropic three-dimensional $\mathrm{XY}$ model. It is assumed that the externally applied magnetic field is low enough to invalidate the lowest Landau-level approach to the problem. A finite-field counterpart of an Onsager vortex-loop transition in extreme type-II superconductors renders the vortex liquid phase incoherent when the Abrikosov vortex lattice undergoes a first-order melting transition. For the magnetic fields considered in this paper, corresponding to filling fractions $f$ given by $1/f=12,14,16,20,25,32,48,64,72,84,96,112,$ and $128,$ the vortex liquid phase is not describable as a liquid of well-defined field-induced vortex lines. This is due to the proliferation of thermally induced closed vortex loops with diameters of the order of the magnetic length in the problem, resulting in a “percolation transition” driven by non-field-induced vortices also transverse to the direction of the applied magnetic field. This immediately triggers flux-line lattice melting and loss of phase coherence along the direction of the magnetic field. Due to this mechanism, the field-induced flux lines lose their line tension in the liquid phase, and cannot be considered to be directed or well defined. In a nonrelativistic two-dimensional boson-analogy picture, this latter feature would correspond to a vanishing mass of the bosons. Scaling functions for the specific heat are calculated in zero and finite magnetic field. From this we conclude that the critical region is of order of $10\%$ of $T_c$ for a mass anisotropy $\sqrt{M_z/M}=3,$ and increases with increasing mass anisotropy. The entropy jump at the melting transition is calculated in two ways as a function of magnetic field for a mass anisotropy slightly lower than that in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ (YBCO), namely, with and without a $T$-dependent prefactor in the Hamiltonian originating at the microscopic level and surfacing in coarse-grained theories such as the one considered in this paper. In the first case, it is found to be $\Delta {S=0.1k}_B$ per pancake vortex, roughly independent of the magnetic field for the filling fractions considered here. In the second case, we find an enhancement of $\Delta S$ by a factor that is less than $2,$ increasing slightly with decreasing magnetic field. This is still lower than experimental values of $\Delta S\approx {0.4k}_B$ found experimentally for YBCO using calorimetric methods. We attribute this to the slightly lower mass anisotropy used in our simulations.