Thermodynamic curvature for a two-parameter spin model with frustration

Microscopic models of realistic thermodynamic systems usually involve a number of parameters, not all of equal macroscopic relevance. We examine a decorated $(1+3)$ Ising spin chain containing two microscopic parameters: a "stiff" $K$ mediating the long-range interactions, and a "sloppy" $J$ operating within local spin groups. $K$ dominates the macroscopic behavior, and varying $J$ has weak effect except in regions where $J$ brings about transitions between phases through its conditioning of the local spin groups with which $K$ interacts. We calculate the heat capacity $C_H$, the magnetic susceptibility $\chi_T$, and the thermodynamic curvature $R$. For large $|J/K|$, we identify four magnetic phases: ferromagnetic, antiferromagnetic, and two ferrimagnetic ones, according to the signs of $K$ and $J$. We argue that for characterizing these phases, the strongest picture is offered by the thermodynamic geometric invariant $R$, proportional to the correlation length $\xi$. This picture has correspondences to other cases, such as fluids.