  Many-body localization provides a generic mechanism of ergodicity breaking in
quantum systems. In contrast to conventional ergodic systems, many-body
localized (MBL) systems are characterized by extensively many local integrals
of motion (LIOM), which underlie the absence of transport and thermalization in
these systems. Here we report a physically motivated construction of local
integrals of motion in the MBL phase. We show that any local operator (e.g., a
local particle number or a spin flip operator), evolved with the system's
Hamiltonian and averaged over time, becomes a LIOM in the MBL phase. Such
operators have a clear physical meaning, describing the response of the MBL
system to a local perturbation. In particular, when a local operator represents
a density of some globally conserved quantity, the corresponding LIOM describes
how this conserved quantity propagates through the MBL phase. Being uniquely
defined and experimentally measurable, these LIOMs provide a natural tool for
characterizing the properties of the MBL phase, both in experiments and
numerical simulations. We demonstrate the latter by numerically constructing an
extensive set of LIOMs in the MBL phase of a disordered spin chain model. We
show that the resulting LIOMs are quasi-local, and use their decay to extract
the localization length and establish the location of the transition between
the MBL and ergodic phases.
