  In recent experiments with ultracold atoms, both two-dimensional (2d) Chern
insulators and one-dimensional (1d) topological charge pumps have been
realized. Without interactions, both systems can be described by the same
Hamiltonian, when some variables are being reinterpreted. In this paper, we
study the relation of both models when Hubbard interactions are added, using
the density-matrix renormalization-group algorithm. To this end, we express the
fermionic Hofstadter model in a hybrid-space representation, and define a
family of interactions, which connects 1d Hubbard charge pumps to 2d Hubbard
Chern insulators. We study a three-band model at particle density $\rho=2/3$,
where the topological quantization of the 1d charge pump changes from Chern
number $C=2$ to $C=-1$ as the interaction strength increases. We find that the
$C=-1$ phase is robust when varying the interaction terms on narrow-width
cylinders. However, this phase does not extend to the limit of the 2d
Hofstadter-Hubbard model, which remains in the $C=2$ phase. We discuss the
existence of both topological phases for the largest cylinder circumferences
that we can access numerically. We note the appearance of a ferromagnetic
ground state between the strongly interacting 1d and 2d models. For this
ferromagnetic state, one can understand the $C=-1$ phase from a bandstructure
argument. Our method for measuring the Hall conductivity could similarly be
realized in experiments: We compute the current response to a weak, linear
potential, which is applied adiabatically. The Hall conductivity converges to
integer-quantized values for large system sizes, corresponding to the system's
Chern number.
