  With rapid progress in control and manipulation of ultracold magnetic atoms
and dipolar molecules, the quantum simulation of lattice models with strongly
interacting dipole-dipole interactions (DDI) and high densities is now within
experimental reach. This rapid development raises the issue about the validity
of quantum simulation in such regimes. In this study, we address this question
by performing a full quantitative comparison between the continuum description
of a one-dimensional gas of dipolar bosons in an optical lattice, and the
single-band Bose-Hubbard lattice model that it quantum simulates. By comparing
energies and density distributions, and by calculating direct overlaps between
the continuum and lattice many-body wavefunctions, we demonstrate that in
regimes of strong DDI and high densities the continuum system fails to recreate
the desired lattice model. Two-band Hubbard models become necessary to reduce
the discrepancy observed between continuum and lattice descriptions, but
appreciable deviations in the density profile still remain. Our study
elucidates the role of strong DDI in generating physics beyond lowest-band
descriptions and should offer a guideline for the calibration of near-term
dipolar quantum simulators.
