Constructing local integrals of motion in the many-body localized phase

We consider a many-body localized spin system and its description by the so-called l-bit Hamiltonian. We outline a renormalization flow procedure to construct the extensive set of conserved quantities, and demonstrate that their locality results in exponentially decaying interactions in this effective model. The associated localization length of this decay is shown to manifest properties very similar to the noninteracting case of Anderson localization: normality of its distribution across samples, and its direct qualitative correspondence to the local spectral properties. A numerical simulation of a magnetic spin-echo protocol quantitatively reproduces these theoretically computed length scales. We therefore argue that these local integrals of motion help to practically identify the many-body localized phase.