RESUMO
In the presence of crystalline symmetries, topological phases of matter acquire a host of invariants leading to nontrivial quantized responses. Here we study a particular invariant, the discrete shift ð®, for the square lattice Hofstadter model of free fermions. ð® is associated with a Z_{M} classification in the presence of M-fold rotational symmetry and charge conservation. ð® gives quantized contributions to (i) the fractional charge bound to a lattice disclination and (ii) the angular momentum of the ground state with an additional, symmetrically inserted magnetic flux. ð® forms its own "Hofstadter butterfly," which we numerically compute, refining the usual phase diagram of the Hofstadter model. We propose an empirical formula for ð® in terms of density and flux per plaquette for the Hofstadter bands, and we derive a number of general constraints. We show that bands with the same Chern number may have different values of ð®, although odd and even Chern number bands always have half-integer and integer values of ð®, respectively.
