Fractional disclination charge and discrete shift in the Hofstadter butterfly

• URL: http://arxiv.org/abs/2204.05320v4
• Date: Sat, 31 Dec 2022 20:57:29 GMT
• Title: Fractional disclination charge and discrete shift in the Hofstadter butterfly
• Authors: Yuxuan Zhang, Naren Manjunath, Gautam Nambiar and Maissam Barkeshli
• Abstract summary: We numerically compute the discrete shift $mathscrS$ for the square lattice Hofstadter model of free fermions. We show that bands with the same Chern number may have different values of $mathscrS$, although odd and even Chern number bands always have half-integer and integer values of $mathscrS$ respectively.
• Score: 15.3862808585761
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\mathscr{S}$, 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.

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