Topological Phase Transitions of Interacting Phases in Commensurate Magnetic Flux*

Lattice Hamiltonians in external magnetic fields provide a non-trivial magnetic translation algebra which results in Lieb-Schultz-Mattis (LSM) type theorems. The LSM theorems impose constraints on the topology of the system, in particuar on its Hall conductivity, and exclude trivial band insulating phases depending on the filling factor. We examine these constraints by taking into account the role of interaction driven spontaneous symmetry breaking of translation symmetry. Using exact diagonalization, we identify phase transitions from Hall insulating to topologically trivial charge density wave states for various flux quantum ratios and filling factors. Our findings demonstrate the importance of "conventional" phase transitions in the study of topological phases as they may provide loopholes for properties otherwise protected by no-go LSM-type theorems.