Thermodynamic geometry of three coupled Ising spins: exact curvature, frustration and bond anisotropy

Abstract

Thermodynamic curvature is widely used as a diagnostic of interactions and phase transitions, yet its interpretation is usually tested only in the thermodynamic limit. Small spin clusters are physically relevant in their own right, since a macroscopic sample of independent molecular trimers or a few-qubit quantum simulator is governed exactly by the few-spin partition function. They also provide the only setting in which the geometry can be obtained in closed form together with its full dependence on the interaction topology. We study the Fisher-Rao (Ruppeiner) geometry of three Ising spins with arbitrary pairwise couplings in a longitudinal field. This cluster interpolates continuously between an open chain and a geometrically frustrated triangle. For the symmetric triangle we obtain the scalar curvature in closed form. It vanishes on the noninteracting line and on a line of fixed field-to-temperature ratio that does not depend on the coupling, which coincides with the inflection point of the single-spin susceptibility. The near-zero behaviour is therefore exactly linear and reflects analyticity rather than critical scaling. In the frustrated antiferromagnetic regime the curvature remains finite and negative down to zero temperature. A general low-temperature formula relates this plateau to the change in magnetization content between the degenerate ground manifold and its first excitation. Any bond anisotropy destroys the plateau below a crossover temperature that depends only logarithmically on the anisotropy, so the curvature is an extremely sensitive detector of exact ground-state degeneracy. On the ferromagnetic side the zero-temperature curvature is universal for all couplings, whereas additional zero lines in anisotropic clusters track field-induced crossings of excited levels.