State space geometry is obtained for the one dimensional Blume Emery Griffiths model and the associated scalar curvature(s) investigated for various parameter regimes, including the Blume-Capel limit and the Griffiths model limit. For the one-dimensional case two complementary geometries with their associated curvatures R m and R q are found which are related to the fluctuations in the two order parameters, namely the magnetic moment and the quadrupole moment. An excellent agreement is obtained in significant regions of the parameter space between the two curvatures and the two corresponding correlation lengths ξ 1 and ξ 2 . The three dimensional scalar curvature R g is also found to efficiently encode interactions. The scaling function for the free energy near critical points and the tricritical point is obtained by making use of Ruppeiner's conjecture relating the inverse of the singular free energy to the thermodynamic scalar curvature.

Four-dimensional state space geometry is worked out for the exactly solved one-dimensional spin-3/2 lattice with a Blume-Emery-Griffiths (BEG) Hamiltonian as well as a more general one with a term containing a non-zero field coupling to the octopole moments. The phase behaviour of the spin-3/2 chain is also explored extensively and novel phenomena suggesting anomalies in the hyperscaling relation and in the decay of fluctuations are reported for a range of parameter values. Using the method of constrained fluctuations worked out earlier in [1, 2] three sectional curvatures and a 3d curvature are obtained and shown to separately encode dipolar, quadrupolar and octopolar correlations both near and away from pseudo-criticality. In all instances of a seeming hyperscaling violation the 3d scalar curvature is found to encode the correlation length while the relevant 2d curvature equals the inverse of singular free energy. For parameter values where the order parameter fluctuation anomalously decays despite a divergence in correlation length the relevant scalar curvature undergoes a sign change to positive values, signalling a possible change in statistics.

We continue our study of the thermodynamic geometry of the spin one model from [1] (paper I) by probing the state space geometry of the Blume Emery Griffiths (BEG) model, and its limiting case of the Blume Capel model, in their mean field approximation. By accounting for the stochastic variables involved we construct from the thermodynamic state space two complimentary two-dimensional geometries with curvatures R m and R q which are shown to encode correlations in the model's two order parameters, namely, the magnetization m and the quadrupole moment q. The geometry is investigated in the zero as well as the non zero 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 Ornstein-Zernicke type correlation length vis-a-vis their scaling properties.

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