  We explore the ground states of strongly interacting bosons in the
vanishingly small and weak lattices using the multiconfiguration time-dependent
Hartree method for bosons (MCTDHB) which calculate numerically exact many-body
wave function. Two new many-body phases: fragmented or quasi superfluid (QSF)
and incomplete fragmented Mott or quasi Mott insulator (QMI) are emerged due to
the strong interplay between interaction and lattice depth. Fragmentation is
utilized as a figure of merit to distinguish these two new phases. We utilize
the eigenvalues of the reduced one-body density matrix and define an order
parameter that characterizes the pathway from a very weak lattice to a deep
lattice. We provide a detailed investigation through the measures of one- and
two-body correlations and information entropy. We find that the structures in
one- and two-body coherence are good markers to understand the gradual built-up
of intra-well correlation and decay of inter-well correlation with increase in
lattice depth.
