Local Sources of Phase Curvature and Rigidity in Finite Quantum Matter

TL;DR

This work reveals that finite coherent quantum systems exhibit a geometric phase rigidity, where localized perturbations generate phase curvature in the many-body state and cause an anomalous breakdown of global phase rigidity. By deriving a quadratic stiffness functional from a Hubbard–Peierls ring and introducing a phase-rigidity scale $\mathcal{R}$, the authors show that $\mathcal{R}$ is governed by the local bond susceptibility $\chi_{\rm loc}$ and can be enhanced by electronic interactions, decoupling from the charge gap $\Delta_{\rm ch}$. Using exact diagonalization, they demonstrate that $\chi_{\rm loc}$ is suppressed with increasing $U$, leading to stronger phase rigidity, while rigidity loss does not require gap formation. The results establish phase rigidity as a geometric diagnostic of coherence protection in finite fermionic matter and suggest experimental platforms in molecular pi-electron rings and mesoscopic quantum circuits to probe this geometry-driven coherence.

Abstract

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.

Figures (2)

• Figure 1: Local sources of phase curvature and definition of the rigidity scale. (a) A finite Hubbard--Peierls ring at half filling subject to a localized bond modulation $t_j=t_0(1+\delta)$, which acts as a local symmetry-breaking source. (b) The local susceptibility $\chi_{\mathrm{loc}}$ is extracted from the curvature of the many-body ground-state energy under the bond modulation, $\chi_{\mathrm{loc}}\equiv -\partial_\delta^2 E_0\!\mid_{\delta=0}$. (c) Within the quadratic effective functional $\Delta E[\{\delta\}]\simeq \frac{1}{2}\sum_{i,j}\delta_i\,\mathcal{K}_{ij}\,\delta_j$, the diagonal element $\mathcal{K}_{jj}=K-\alpha^2 t_0^{\,2}\chi_{\mathrm{loc}}\equiv\mathcal{R}$ defines the local phase-rigidity scale.
• Figure 2: Local susceptibility does not track the charge gap. The local susceptibility $\chi_{\mathrm{loc}}(U)\equiv -\partial_\delta^2 E_0\!\mid_{\delta=0}$ (left axis, logarithmic scale), extracted from the curvature of the ground-state energy under a single-bond modulation, is reported together with the charge gap $\Delta_{\mathrm{ch}}(U)$ (right axis) for a Hubbard ring with $N=8$ sites at half filling. While $\Delta_{\mathrm{ch}}(U)$ increases monotonically with interaction strength, $\chi_{\mathrm{loc}}(U)$ is sharply reduced for $U>0$ and then varies only weakly (or decreases) across the correlated regime. Thus, the local susceptibility does not track the gap, showing that coherence protection is not a trivial consequence of spectral gapping.