  Eigenstates of many-body localised (MBL) systems are characterised by
area-law bipartite entanglement along with multifractal statistics of their
amplitudes on Hilbert space. At the same time, sparse random pure states with
fractal statistics are not compatible with area-law and necessarily exhibit
volume-law entanglement. This raises the question that what correlation
functions of Hilbert-space amplitudes MBL eigenstates must possess for their
area law entanglement to be compatible with their multifractality. In this
work, we identify and compute such appropriate Hilbert-space correlations which
carry information of entanglement. We find that, for MBL eigenstates, these
correlations are qualitatively different not only from those of ergodic states
but also of sparse random states with fractal statistics. This enables us to
show that indeed the said correlations lie at the heart of the coexistence of
area-law entanglement and multifractality for MBL eigenstates.

Disordered Systems and Neural Networks;Statistical Mechanics;Strongly Correlated Electrons;Quantum Physics