  At the familiar liquid-gas phase transition in water, the density jumps
discontinuously at atmospheric pressure, but the line of these first-order
transitions defined by increasing pressures terminates at the critical point, a
concept ubiquitous in statistical thermodynamics. In correlated quantum
materials, a critical point was predicted and measured terminating the line of
Mott metal-insulator transitions, which are also first-order with a
discontinuous charge density. In quantum spin systems, continuous quantum phase
transitions (QPTs) have been investigated extensively, but discontinuous QPTs
have received less attention. The frustrated quantum antiferromagnet
SrCu$_2$(BO$_3$)$_2$ constitutes a near-exact realization of the paradigmatic
Shastry-Sutherland model and displays exotic phenomena including magnetization
plateaux, anomalous thermodynamics and discontinuous QPTs. We demonstrate by
high-precision specific-heat measurements under pressure and applied magnetic
field that, like water, the pressure-temperature phase diagram of
SrCu$_2$(BO$_3$)$_2$ has an Ising critical point terminating a first-order
transition line, which separates phases with different densities of magnetic
particles (triplets). We achieve a quantitative explanation of our data by
detailed numerical calculations using newly-developed finite-temperature
tensor-network methods. These results open a new dimension in understanding the
thermodynamics of quantum magnetic materials, where the anisotropic spin
interactions producing topological properties for spintronic applications drive
an increasing focus on first-order QPTs.
