In the presence of crystalline symmetries, topological phases of matter acquire a host of invariants leading to non-trivial quantized responses. Here we study a particular invariant, the discrete shift $\mathscr{ S}$, for the square lattice Hofstadter model of free fermions. $\mathscr{ S}$ is associated with a $\mathbb{ Z}_M$ classification in the presence of $M$-fold rotational symmetry and charge conservation. $\mathscr{ S}$ 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. $\mathscr{ S}$ forms its own `Hofstadter butterfly', which we numerically compute, refining the usual phase diagram of the Hofstadter model. We propose an empirical formula for $\mathscr{ S}$ 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 $\mathscr{ S}$, although odd and even Chern number bands always have half-integer and integer values of $\mathscr{ S}$ respectively.
Categories: cond-mat.str-el cond-mat.mes-hall quant-ph