We prove localization for the random XXZ quantum spin chain in the
multi-cluster spectrum. The fractional moment method (FMM) is performed
directly on the XXZ quantum spin chain Hamiltonian, using tools suitable for
spin models. This requires significant modifications of the usual FMM method
for random Schr\”odinger operators. In particular, the argument relies on a
suitable replacement for the Green’s function, the subject of the usual FMM.
Performing the FMM on this new quantity requires a new set of ideas.
