Relating different localization lengths via non-perturbative construction of local integrals of motion in the many-body localized phase

Many-body localization (MBL) is characterized by the absence of thermalization and the violation of conventional thermodynamics. The phenomenological model, which describes the system using a complete set of local integrals of motion (LIOMs), provides a powerful tool to understand MBL. We explicitly compute a complete set of LIOMs non-perturbatively by maximizing the overlap between LIOMs and physical spin operators. This method enables a direct mapping from real space Hamiltonian to the phenomenological model. We demonstrate the exponential decay of weight of LIOMs in real-space and interaction strength of LIOMs in range. We further compare the localization lengths extracted from LIOMs, their interactions and dynamics. Our scheme is immune to accidental resonances and can be applied even at the phase transition point, providing a novel tool to study the microscopic features of the phenomenological model of MBL.