  Adding magnetic flux to a band structure breaks Bloch's theorem by realizing
a projective representation of the translation group. The resulting Hofstadter
spectrum encodes the non-perturbative response of the bands to flux. Depending
on their topology, adding flux can enforce a bulk gap closing (a Hofstadter
semimetal) or boundary state pumping (a Hofstadter topological insulator). In
this work, we present a real-space classification of these Hofstadter phases.
We give topological indices in terms of symmetry-protected Real Space
Invariants (RSIs) which encode bulk and boundary responses of fragile
topological states to flux. In fact, we find that the flux periodicity in
tight-binding models causes the symmetries which are broken by the magnetic
field to reenter at strong flux where they form projective point group
representations. We completely classify the reentrant projective point groups
and find that the Schur multipliers which define them are Arahanov-Bohm phases
calculated along the bonds of the crystal. We find that a nontrivial Schur
multiplier is enough to predict and protect the Hofstadter response with only
zero-flux topology.
