Hofstadter Topology: Complete Classification and Non-crystalline Projective Symmetries

The Hofstadter problem is the lattice analog of the quantum Hall effect and is the paradigmatic example of topology induced by an applied magnetic field. Conventionally, the Hofstadter problem involves adding approximately 10⁴ T magnetic fields to a trivial band structure. In this work, we show that when a magnetic field is added to an initially topological band structure, a wealth of possible phases emerges. First, we prove that at fixed filling, a nonzero Chern number creates a discontinuous many-body gap closing at zero flux, and that a nonzero mirror or valley Chern number enforces a bulk gap closing at finite flux. We then study Hofstadter Hamiltonians with nontrivial space group symmetries. Remarkably, we find that at critical values of the flux, the symmetries realize projective representations of the space group which cannot be obtained in any crystalline insulator. We completely classify these novel space groups and their topological invariants, and show that they are be determined by the zero-flux Wannier functions. Our classification reveals topologically protected edge and corner mode pumping, which we expect to be observable in Moiré materials where laboratory-strength fields can reach one flux per unit cell.