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

Figure 1

$𝒮$ for Hofstadter model, from Eq. (5).

Figure 2

(a) Lattice disclination with disclination angle $\mathrm{\Omega }=(\pi /2)$. The blue region $W$ covers 11 unit cells. $Q_i$ is weighted by the indicated amount when calculating $Q_W$ or ${\overline{Q}}_W$. (b) Standard deviation of $𝒮$ as a function of bond disorder ${\sigma }_{\mathrm{bond}}$ or on site disorder ${\sigma }_{\text{on site}}$ for $C=1$ and $C=2$ main Landau level (average hopping is one).

Figure 3

(a)–(c) $Q_i$ for each site $i$ in a clean system with parameters (a) $(\varphi /2\pi )=\epsilon$, $C=1$, (b) $(\varphi /2\pi )=\frac{1}{2}-\epsilon$, $C=-2$, (c) $(\varphi /2\pi )=\frac{1}{3}+\epsilon$, $C=3$. The color bar is in units of ${\nu }_0$. $\epsilon$ is a small fraction that opens up the band gap. (d) ${\overline{Q}}_W$ for (a)–(c) is quantized at $(𝒮/4)$ when $R\ge 9$. $R$ is the distance from the disclination center to $\partial W$ (total side length is $L=24$ unit cells). Here we have cropped out the edges to show the bulk features more clearly; the full figures with edges are given in the Supplemental Material [35], Sec. V.