High-magnetic field magnetostriction and thermal expansion in URu$_2$Si$_2$

We report high magnetic field (up to $\mu_0$H = 45 T) $c$-axis thermal expansion and magnetostriction experiments on URu$_2$Si$_2$ single crystals. The sample length change $\Delta$L$_c$(T$_{HO}$)/L$_c$ associated with the transition to the ``hidden order'' phase becomes increasingly discontinous as the magnetic field is raised above 25 T. The re-entrant ordered phase III is clearly observed in both the thermal expansion $\Delta$L$_c$(T)/L$_c$ and magnetostriction $\Delta$L$_c$(B)/L$_c$ above 36 T, in good agreement with previous results. The sample length is also discontinous at the boundaries of this phase, mainly at the upper boundary. A change in the sign of the coefficient of thermal-expansion is observed at the metamagnetic transition (B$_M$ ? 38 T) which is likely related to the existence of a quantum critical end point. See V.F. Correa et al., Phys. Rev. Lett. (in the press).