Many-body geometric singularities without quantum criticality

Geometric approaches to thermodynamics and quantum statistical mechanics have established the divergence of the Ruppeiner curvature as a widely used indicator of critical behavior and long-range correlations. In this work we present an explicit many-body equilibrium model in which a divergent thermodynamic curvature emerges in the complete absence of quantum criticality. Focusing on an interacting spin-flip SU(2) fermionic system at finite temperature, we show that increasing interaction strength drives the Gibbs state toward a pure configuration through an interaction-induced narrowing of the accessible Hilbert space. In this regime, the Fisher information with respect to the interaction parameter vanishes, signaling statistical rigidity rather than enhanced fluctuations, while the Ruppeiner curvature diverges due to the collapse of the underlying statistical manifold. We demonstrate that this curvature divergence originates from geometric degeneracy rather than from critical correlations, providing a concrete counterexample to the common identification of geometric singularities with quantum phase transitions. Our results highlight the necessity of jointly analyzing curvature, Fisher information, and state purity when using information-geometric diagnostics to characterize many-body systems.