Geometric phase and multipartite entanglement of Rydberg atom chains

We investigate the behavior of geometric phase (GP) and geometric entanglement (GE), a multipartite entanglement measure, across quantum phase transitions in Rydberg atom chains. Using density-matrix renormalization-group calculations and finite-size scaling analysis, we characterize the critical properties of transitions between disordered and ordered phases. Both quantities exhibit characteristic scaling near transition points, with the disorder to the ${Z}_{2}$ ordered phase transition showing behavior consistent with the Ising universality class, while the disorder to the ${Z}_{3}$ phase transition displays distinct critical properties. We demonstrate that GP and GE serve as sensitive probes of quantum criticality, providing consistent critical parameters and scaling behavior. A unifying description of these geometric quantities from a quantum geometry perspective is explored, and an interferometric setup for their potential measurement is discussed. Our results provide insights into the interplay between geometric phase and multipartite entanglement near quantum phase transitions in Rydberg-atom systems, revealing how these quantities reflect the underlying critical behavior in these complex quantum many-body systems.