  The interacting dark energy (IDE) model, which considers the interaction
between dark energy and dark matter, provides a natural mechanism to alleviate
the coincidence problem and can also relieve the observational tensions under
the $\Lambda$CDM model. Previous studies have put constraints on IDE models by
observations of cosmic expansion history, cosmic microwave background and
large-scale structures. However, these data are not yet enough to distinguish
IDE models from $\Lambda$CDM effectively. Because the non-linear structure
formation contains rich cosmological information, it can provide additional
means to differentiate alternative models. In this paper, based on a set of
$N$-body simulations for IDE models, we investigate the formation histories and
properties of dark matter halos, and compare with their $\Lambda$CDM
counterparts. For the model with dark matter decaying into dark energy and the
parameters being the best-fit values from previous constraints, the structure
formation is markedly slowed down, and the halos have systematically lower
mass, looser internal structure, higher spin and anisotropy. This is
inconsistent with the observed structure formation, and thus this model can be
safely ruled out from the perspective of non-linear structure formation.
Moreover, we find that the ratio of halo concentrations between IDE and
$\Lambda$CDM counterparts depends sensitively on the interaction parameter and
is independent of halo mass. This can act as a powerful probe to constrain IDE
models. Our results concretely demonstrate that the interaction of the two dark
components can affect the halo formation considerably, and therefore the
constraints from non-linear structures are indispensable.
