Investigation of spin-1 Blume-Emery-Griffiths model in theframework of Riemannian geometry

A method combining statistical equilibrium theory and Riemannian metric geometry is used to investigate the properties of thermodynamic curvature or Ricci scalar (R) for the isotropic and anisotropic spin-1 Blume-Emery-Griffiths (BEG) models. A Ruppeiner metric is introduced on the two-dimensional phase space of dipolar and quadrupolar order parameters. An expression is derived for R and its variation as a function of reduced temperature (θ) and reduced cyrstal field (d) for various coupling ratio (α; r) particularly its behaviors near the first-order and second-order phase transitions and critical/tricritical/multicritical points is examined numerically. R tends towards plus infinity while approaching the second order phase transition and tricritical point. These results fit well with those in the mean-field Ising model in the lowest order approximation and quantum lattice model with multi-well potentials. Finally, geometric phase diagrams, including critical/multicritical topology, are presented in the (θ; α) and (d; θ) planes for isotropic and anisotropic spin-1 BEG models, respectively.