  Studies of disordered spin chains have recently experienced a renewed
interest, inspired by the question to which extent the exact numerical
calculations comply with the existence of a many-body localization phase
transition. For the paradigmatic random field Heisenberg spin chains, many
intriguing features were observed when the disorder is considerable compared to
the spin interaction strength. Here, we introduce a phenomenological theory
that may explain some of those features. The theory is based on the proximity
to the noninteracting limit, in which the system is an Anderson insulator.
Taking the spin imbalance as an exemplary observable, we demonstrate that the
proximity to the local integrals of motion of the Anderson insulator determines
the dynamics of the observable at infinite temperature. In finite interacting
systems our theory quantitatively describes its integrated spectral function
for a wide range of disorders.
