We report the results of the numerical study of the non-dissipative quantum
Josephson junction chain with the focus on the statistics of many-body wave
functions and local energy spectra. The disorder in this chain is due to the
random offset charges. This chain is one of the simplest physical systems to
study many-body localization. We show that the system may exhibit three
distinct regimes: insulating, characterized by the full localization of
many-body wavefunctions, fully delocalized (metallic) one characterized by the
wavefunctions that take all the available phase volume and the intermediate
regime in which the volume taken by the wavefunction scales as a non-trivial
power of the full Hilbert space volume. In the intermediate, non-ergodic regime
the Thouless conductance (generalized to many-body problem) does not change as
a function of the chain length indicating a failure of the conventional
single-parameter scaling theory of localization transition. The local spectra
in this regime display the fractal structure in the energy space which is
related with the fractal structure of wave functions in the Hilbert space. A
simple theory of fractality of local spectra is proposed and a new scaling
relationship between fractal dimensions in the Hilbert and energy space is
suggested and numerically tested.Comment: 11 page