Response to a local quench of a system near many body localization transition

We consider a one dimensional spin $1/2$ chain with Heisenberg interaction in a disordered parallel magnetic field. This system is known to exhibit the many body localization (MBL) transition at critical strength of disorder. We analyze the response of the chain when additional perpendicular magnetic field is applied to an individual spin and propose a method for accurate determination of the mobility edge via local spin measurements. We further demonstrate that the exponential decrease of the spin response with the distance between perturbed spin and measured spin can be used to characterize the localization length in the MBL phase.