  Motivated by recently reported magnetic-field induced topological phases in
ultracold atoms and correlated Moir\'e materials, we investigate topological
phase transitions in a minimal model consisting of interacting spinless
fermions described by the Hofstadter model on a square lattice. For interacting
lattice Hamiltonians in the presence of a commensurate magnetic flux it has
been demonstrated that the quantized Hall conductivity is constrained by a
Lieb-Schultz-Mattis (LSM)-type theorem due to magnetic translation symmetry. In
this work, we revisit the validity of the theorem for such models and establish
that a topological phase transition from a topological to a trivial insulating
phase can be realized but must be accompanied by spontaneous magnetic
translation symmetry breaking caused by charge ordering of the spinless
fermions. To support our findings, the topological phase diagram for varying
interaction strength is mapped out numerically with exact diagonalization for
different flux quantum ratios and band fillings using symmetry indicators. We
discuss our results in the context of the LSM-type theorem.
