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Fractional Disclination Charge and Discrete Shift in the Hofstadter Butterfly.

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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 ๐’ฎ,… Click to show full abstract

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 ๐’ฎ, for the square lattice Hofstadter model of free fermions. ๐’ฎ is associated with a Z_{M} classification in the presence of M-fold rotational symmetry and charge conservation. ๐’ฎ gives quantized contributions to (i)ย the fractional charge bound to a lattice disclination and (ii)ย the angular momentum of the ground state with an additional, symmetrically inserted magnetic flux. ๐’ฎ forms its own "Hofstadter butterfly," which we numerically compute, refining the usual phase diagram of the Hofstadter model. We propose an empirical formula for ๐’ฎ 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 ๐’ฎ, although odd and even Chern number bands always have half-integer and integer values of ๐’ฎ, respectively.

Keywords: disclination; discrete shift; charge; hofstadter; hofstadter butterfly

Journal Title: Physical review letters
Year Published: 2022

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