One of the most fascinating properties of high-T-c superconductors is
the existence of a thermodynamic phase-transition located well below t
he mean field transition line H-c2. This transition is a first order m
elting transition of the Abrikosov vortex lattice into a entangled vor
tex liquid, and is signalled by a discontinuity in the magnetization.
On the London level, the vortex system can be mapped onto interacting
bosons in two dimensions. We use this mapping to simulate the vortex s
ystem with Path Integral Monte Carlo methods. This approach does not i
ntroduce an underlying lattice and allows for the simultaneous investi
gation of crystalline and longitudinal long-range order. We find a fir
st-order transition as the system passes from a lattice to an entangle
d liquid. The jump in the magnetization is in agreement with experimen
ts. The phase diagrams of bosons and of vortices are discussed.