A rigorous proof of integrability or nonintegrability in quantum many-body systems is among the most challenging tasks, as it involves demonstrating the presence or absence of local conserved quantities and deciphering the complex dynamics of the system. In this paper, we establish a graph-theoretical analysis as a comprehensive framework for proving the nonintegrability of quantum systems. Exemplifying the PXP model, which is widely believed to be nonintegrable, this paper rigorously proves the absence of local conserved quantities, thereby confirming its nonintegrability. This proof for the PXP model gives several important messages not only that the system is nonintegrable but also the quantum many-body scarring observed in the model is not associated with the existence of local conserved quantities. From a graph-theoretical perspective, we also highlight its advantage, even in integrable systems, as the classification of local conserved quantities can be achieved by simply counting the number of isolated loops in the graphs. Our approach is broadly applicable for establishing proofs of (non)integrability in other quantum many-body systems, significantly simplifying the process of proving nonintegrability and giving numerous potential applications.

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• Integrability in field theory
• Quantum scars
• Quantum many-body systems
• Rydberg atoms & molecules