Emergent local integrals of motion without a complete set of localized eigenstates

Systems where all energy eigenstates are localized are known to display an emergent local integrability, in the sense that one can construct an extensive number of operators that commute with the Hamiltonian and are localized in real space. Here we show that emergent local integrability does not require a complete set of localized eigenstates. Given a set of localized eigenstates comprising a nonzero fraction $(1-f)$ of the full many body spectrum, one can construct an extensive number of integrals of motion which are local in the sense that they have {\it nonzero weight} in a compact region of real space, in the thermodynamic limit. However, these modified integrals of motion have a `global dressing’ whose weight vanishes as $\sim f$ as $f\to 0$. In this sense, the existence of a {\it non-zero fraction} of localized eigenstates is sufficient for emergent local integrability. We discuss the implications of our findings for systems where the spectrum contains delocalized states, for systems with projected Hilbert spaces, and for the robustness of quantum integrability.