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.
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