  We construct a solvable spin chain model of many-body localization (MBL) with
a tunable mobility edge. This simple model not only demonstrates analytically
the existence of mobility edges in interacting one-dimensional (1D) disordered
systems, but also allows us to study their physics. By establishing a
connection between MBL and a quantum central limit theorem (QCLT), we show that
many-body localization-delocalization transitions can be visualized as tuning a
mobility edge in the energy spectrum. Since the effective disorder strength for
individual eigenstates depends on energy density, we identify "energy-resolved
disorder strength" as a physical mechanism for the appearance of mobility
edges, and support the universality of this mechanism by arguing its presence
in a large class of models including the random-field Heisenberg chain. We also
construct models with multiple mobility edges. All our constructions can be
made translationally invariant.
