Local Sources of Phase Curvature and Rigidity in Finite Quantum Matter
Finite coherent quantum systems exhibit a nontrivial response to local sources of phase curvature, which cannot be reduced to conventional forces, disorder-induced localization, or simple gap opening. Here we show that, in finite fermionic rings, a localized symmetry-breaking perturbation acts as a source of phase curvature in the many-body Hilbert space, inducing an anomalous breakdown of global phase rigidity. Starting from a Hubbard-Peierls description, we derive an effective field-theoretic functional in which the inverse local susceptibility defines a phase-rigidity scale controlled by system size and electronic correlations. This rigidity quantifies the resistance of a coherent many-body state to geometric deformation of its phase structure, rather than to energetic localization. We demonstrate that interactions enhance phase rigidity in finite systems, counter to naive expectations based on single-particle localization, and that rigidity loss may occur without a direct correspondence to gap formation. Molecular pi-electron rings and mesoscopic quantum circuits provide experimentally accessible realizations of this regime, establishing a direct connection between local phase curvature, geometric rigidity, and coherence-driven phenomena across finite quantum matter.