Fractional disclination charge and discrete shift in the Hofstadter butterfly

Abstract

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 $\mathcal{S}$, for the square lattice Hofstadter model of free fermions. $\mathcal{S}$ is associated with a ${\mathbb{Z}}_M$ classification in the presence of $M$-fold rotational symmetry and charge conservation. $\mathcal{S}$ 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. $\mathcal{S}$ forms its own `Hofstadter butterfly', which we numerically compute, refining the usual phase diagram of the Hofstadter model. We propose an empirical formula for $\mathcal{S}$ 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 $\mathcal{S}$, although odd and even Chern number bands always have half-integer and integer values of $\mathcal{S}$ respectively.

arXiv comment6+14 pages. Minor edits to the main text and the appendix

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