Nonintegrability in the PXP model: A graph-theoretical approach

Establishing a rigorous proof of integrability or nonintegrability in quantum many-body systems remains one of the most challenging tasks, as it requires demonstrating the presence or absence of local conserved quantities and deciphering the complex dynamics of the system. In this paper, we introduce a graph-theoretical approach to address this issue. This framework not only provides a comprehensive method for analyzing the nonintegrability of quantum systems but also brings advantages for integrable systems by enabling the classification of local conserved quantities through a straightforward count of isolated loops in the associated graphs. As an illustrative example, we focus on the PXP model, a system widely believed to be nonintegrable, and rigorously confirm its nonintegrability by demonstrating the absence of local conserved quantities through graph-theoretical analysis. This proof for the PXP model conveys several key insights: It not only confirms the model's nonintegrability but also indicates that the quantum many-body scarring properties of the model is not associated with local conserved quantities. Our innovative approach is broadly applicable for proving (non-)integrability in other quantum many-body systems, significantly simplifying the process and offering numerous potential applications.