A quantum magnetic analogue to the critical point of water

At the liquid–gas phase transition in water, the density has a discontinuity at atmospheric pressure; however, the line of these first-order transitions defined by increasing the applied pressure terminates at the critical point¹, a concept ubiquitous in statistical thermodynamics². In correlated quantum materials, it was predicted³ and then confirmed experimentally^4,5 that a critical point terminates the line of Mott metal–insulator transitions, which are also first-order with a discontinuous charge carrier density. In quantum spin systems, continuous quantum phase transitions⁶ have been controlled by pressure^7,8, applied magnetic field^9,10 and disorder¹¹, but discontinuous quantum phase transitions have received less attention. The geometrically frustrated quantum antiferromagnet SrCu₂(BO₃)₂ constitutes a near-exact realization of the paradigmatic Shastry–Sutherland model^12–14 and displays exotic phenomena including magnetization plateaus¹⁵, low-lying bound-state excitations¹⁶, anomalous thermodynamics¹⁷ and discontinuous quantum phase transitions^18,19. Here we control both the pressure and the magnetic field applied to SrCu₂(BO₃)₂ to provide evidence of critical-point physics in a pure spin system. We use high-precision specific-heat measurements to demonstrate that, as in water, the pressure–temperature phase diagram has a first-order transition line that separates phases with different local magnetic energy densities, and that terminates at an Ising critical point. We provide a quantitative explanation of our data using recently developed finite-temperature tensor-network methods^17,20–22. These results further our understanding of first-order quantum phase transitions in quantum magnetism, with potential applications in materials where anisotropic spin interactions produce the topological properties^23,24 that are useful for spintronic applications.