Proof of the nonintegrability of spin-$1/2$ systems*

We suggest a general framework which can systematically show the nonintegrability of arbitrary Hamiltonian. For spin-$1/2$ systems, we transformed the existence problem of the local conserved quantity into the graph theoretical problem. This transformation helps to reduce lots of the candidates of the local conserved quantity, aiding the proof of nonintegrability of Hamiltonian. As an example, we show the "Quantum Many-Body Scar" PXP model is nonintegrable, which is a nice example because the PXP model contains every possible cases which are hard to treat. We also show that other various models can be shown to be nonintegrable in our framework.