Thermodynamic metric geometry of the two-state ST2 model for supercooled water

Liquid water has anomalous liquid properties, such as its density maximum at 4\degree C. An attempt at theoretical explanation proposes a liquid-liquid phase transition line in the supercooled liquid state, with coexisting low-density (LDL) and high-density (HDL) liquid states. This line terminates at a critical point. It is assumed that the LDL state possesses mesoscopic tetrahedral structures that give it solid-like properties, while the HDL is a regular random liquid. But the short-lived nature of these solid-like structures make them difficult to detect directly. We take a thermodynamic approach instead, and calculate the thermodynamic Ricci curvature scalar $R$ in the metastable liquid regime. It is believed that solid-like structures signal their presence thermodynamically by a positive sign for $R$, with a negative sign typically present in less organized fluid states. Using thermodynamic data from ST2 computer simulations fit to a mean field (MF) two state equation of state, we find significant regimes of positive $R$ in the LDL state, supporting the proposal of solid-like structures in liquid water. In addition, we review the theory, compute critical exponents, demonstrate the large reach of the MF critical regime, and calculate the Widom line using $R$.