  Fermionization is what happens to the state of strongly interacting repulsive
bosons interacting with contact interactions in one spatial dimension.
Crystallization is what happens for sufficiently strongly interacting repulsive
bosons with dipolar interactions in one spatial dimension. Crystallization and
fermionization resemble each other: in both cases -- due to their repulsion --
the bosons try to minimize their spatial overlap. We trace these two hallmark
phases of strongly correlated one-dimensional bosonic systems by exploring
their ground state properties using the one- and two-body density matrix. We
solve the $N$-body Schr\"odinger equation accurately and from first principles
using the multiconfigurational time-dependent Hartree for bosons (MCTDHB) and
for fermions (MCTDHF) methods. Using the one- and two-body density,
fermionization can be distinguished from crystallization in position space. For
$N$ interacting bosons, a splitting into an $N$-fold pattern in the one-body
and two-body density is a unique feature of both, fermionization and
crystallization. We demonstrate that the splitting is incomplete for
fermionized bosons and restricted by the confinement potential. This incomplete
splitting is a consequence of the convergence of the energy in the limit of
infinite repulsion and is in agreement with complementary results that we
obtain for fermions using MCTDHF. For crystalline bosons, in contrast, the
splitting is complete: the interaction energy is capable of overcoming the
confinement potential. Our results suggest that the spreading of the density as
a function of the dipolar interaction strength diverges as a power law. We
describe how to distinguish fermionization from crystallization experimentally
from measurements of the one- and two-body density.
