  Many-body localization (MBL), characterized by the absence of thermalization
and the violation of conventional thermodynamics, has elicited much interest
both as a fundamental physical phenomenon and for practical applications in
quantum information. A phenomenological model, which describes the system using
a complete set of local integrals of motion (LIOMs), provides a powerful tool
to understand MBL, but can be usually only computed approximately. Here we
explicitly compute a complete set of LIOMs with a non-perturbative approach, by
maximizing the overlap between LIOMs and physical spin operators in real space.
The set of LIOMs satisfies the desired exponential decay of weight of LIOMs in
real-space. This LIOM construction enables a direct mapping from the real space
Hamiltonian to the phenomenological model and thus enables studying the
localized Hamiltonian and the system dynamics. We can thus study and compare
the localization lengths extracted from the LIOM weights, their interactions,
and dephasing dynamics, revealing interesting aspects of many-body
localization. Our scheme is immune to accidental resonances and can be applied
even at phase transition point, providing a novel tool to study the microscopic
features of the phenomenological model of MBL.
