Observation of a vortex lattice spinodal and a Bragg glass in MgB$_{2}$

We observed a fully disordered (Spinodal) and a fully ordered (Bragg glass) vortex lattice in MgB$_{2}$ single crystals by using fast current measurements to measure $I_{c}$ for variously prepared vortex states. The obtained $I_{c}$ showed a very different behavior depending on the vortex states, such as Zero Field Cooled (ZFC), ZFCW (warmed), FC, and FCW. We noticed that the values of $I_{c}(T)$ for the FC and the FCW states for MgB$_{2}$ showed clear history dependence with two reversible points. The one point above the $T_{peak}$ was exactly the same as the spinodal point obtained by using the same method as reported in PRL 92, 227004 (2004). The other point below the $T_{peak,}$ was consistent with the Bragg glass point measured by using the time-resolved transport measurements reported in PRL 96, 017009 (2006). We obtained $H-T$ diagram including several vortex states. The theoretical spinodal line predicted by Li \textit{et al. }based on one-gap superconductivity described the case for NbSe$_{2 }$well, but could not explain the spinodal line for MgB$_{2}$.