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

Abstract

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.

7 pages, 5 figures, 3 appendices

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• Droplet localization in the random XXZ model and its manifestations
• Multibody expansion of the local integrals of motion: How many pairs of particle-hole do we really need to describe the quasiparticles in the many-body localized phase?
• Symmetry- and energy-resolved entanglement dynamics in a disordered Bose-Hubbard model
• Critical localization with Van der Waals interactions
• Many-body localization and particle multioccupancy in the disordered Bose-Hubbard model