Fractional disclination charge and discrete shift in the Hofstadter butterfly

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.