Equilibrium First-Order Melting and Second-Order Glass Transitions of the Vortex Matter in Bi${}_2$Sr${}_2$CaCu${}_2$O${}_8$

The thermodynamic $H-T$ phase diagram of Bi${}_2$Sr${}_2$CaCu${}_2$O${}_8$ was mapped by measuring local \emph{equilibrium} magnetization $M(H,T)$ in presence of vortex `shaking’. Two equally sharp first-order magnetization steps are revealed in a single temperature sweep, manifesting a liquid-solid-liquid sequence. In addition, a second-order glass transition line is revealed by a sharp break in the equilibrium $M(T)$ slope. The first- and second-order lines intersect at intermediate temperatures, suggesting the existence of four phases: Bragg glass and vortex crystal at low fields, glass and liquid at higher fields.