  Higher order topological insulators (HOTIs) are a novel form of insulating
quantum matter, which are characterized by having gapped boundaries that are
separated by gapless corner or hinge states. Recently, it has been proposed
that the essential features of a large class of HOTIs are captured by
topological multipolar response theories. In this work, we show that these
multipolar responses can be realized in interacting lattice models, which
conserve both charge and dipole. In this work we study several models in both
the strongly interacting and mean-field limits. In $2$D we consider a
ring-exchange model which exhibits a quadrupole response, and can be tuned to a
$C_4$ symmetric higher order topological phase with half-integer quadrupole
moment, as well as half-integer corner charges. We then extend this model to
develop an analytic description of adiabatic dipole pumping in an interacting
lattice model. The quadrupole moment changes during this pumping process, and
if the process is periodic, we show the total change in the quadrupole moment
is quantized as an integer. We also consider two interacting $3$D lattice
models with chiral hinge modes. We show that the chiral hinge modes are heralds
of a recently proposed "dipolar Chern-Simons" response, which is related to the
quadrupole response by dimensional reduction. Interestingly, we find that in
the mean field limit, both the $2$D and $3$D interacting models we consider
here are equivalent to known models of non-interacting HOTIs (or boundary
obstructed versions). The self-consistent mean-field theory treatment provides
insight into the connection between free-fermion (mean-field) theories having
vanishing polarization and interacting models where dipole moments are
microscopically conserved.
