Thermodynamic geometry of spin-one lattice models. II. Criticality and coexistence in the mean-field approximation

We continue our study of the thermodynamic geometry of the spin one model from A. Sahay and R. Sanwari, Phys. Rev. E 105, 034134 (2022)10.1103/PhysRevE.105.034134. by probing the state space geometry of the Blume-Emery-Griffiths model and the Blume-Capel model in their mean-field approximation. By accounting for the stochastic variables involved we construct from the thermodynamic state space two complementary two-dimensional geometries with curvatures R_{m} and R_{q} which are shown to encode correlations in the model's two order parameters, the magnetization m and the quadrupole moment q. The geometry is investigated in the zero as well as the nonzero magnetic field region. We find that the relevant scalar curvatures diverge to negative infinity along the critical lines with the correct scaling and amplitude. We then probe the geometry of phase coexistence and find that the relevant curvatures predict the coexistence curve remarkably well via their respective R-crossing diagrams. We also briefly comment on the effectiveness of the geometric correlation length compared to the commonly used Ginzburg-Landau correlation length vis-à-vis their scaling properties.