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Inspired by recent experimental measurements [Guo \textit{et al.}, Phys. Rev.Lett.~\textbf{124}, 206602 (2020); Jim\'enez \textit{et al.}, Nature\textbf{592}, 370 (2021)] on frustrated quantum magnet SrCu$_2$(BO$_3$)$_2$under combined pressure and magnetic fields, we study the related spin-$1/2$Shastry-Sutherland (SS) model using state-of-the-art tensor network methods. Bycalculating thermodynamics, correlations and susceptibilities, we find, in zeromagnetic field, not only a line of first-order plaquette-singlet (PS) todimer-singlet phase transition ending with a critical point, but alsosignatures of the ordered PS transition with its critical endpoint terminatingon this first-order line. Moreover, we uncover prominent magnetic barocaloricresponses, a novel type of quantum correlation induced cooling effect, in thestrongly fluctuating supercritical regime. Under finite fields, we identify aquantum phase transition from the PS phase to the spin supersolid phase thatbreaks simultaneously lattice translational and spin rotational symmetries. Thepresent findings on the SS model are accessible in current experiments andwould shed new light on exotic critical and supercritical phenomena inarchetypal frustrated quantum magnets.

Plaquette Singlet Transition, Magnetic Barocaloric Effect, and Spin Supersolidity in the Shastry-Sutherland Model