Interacting systems with zero thermodynamic curvature

We review a conjecture by Ruppeiner that relates the nature of interparticle interactions to the sign of the thermodynamic curvature scalar R, paying special attention to the case of zero curvature. We highlight the underappreciated fact that there are two Ruppeiner metrics, and corresponding curvature scalars, that are equally viable in principle, obtained by restricting to systems of constant volume and constant particle number, respectively. We then demonstrate the existence of thermodynamic systems with vanishing curvature scalar but nontrivial interactions. Information about interactions in these systems is obtained by carrying out an inversion procedure on the virial coefficients. We demonstrate that these nontrivial zero-curvature systems are not always spurious or unphysical, as they include the well-known hard-sphere potential, as well as a system whose virial expansion matches, up to the second virial coefficient, that of a particular inverse-power potential. Finally, we argue that the ideal gas is the unique physical system for which both curvature scalars vanish. This leads us to propose an extension to Ruppeiner's conjecture.