Fig. 2: First-order 0D dislocation states in 2D crystals from 1D polarization topology.

a The bulk Brillouin zone (BZ) of a 2D rectangular magnetic crystal with only \({{{{{{{\mathcal{I}}}}}}}}\) symmetry. b An \({{{{{{{\mathcal{I}}}}}}}}\)-related pair of 0D dislocations with Burgers vector \({{{{{{{\boldsymbol{B}}}}}}}}=\hat{y}\) in an \({{{{{{{\mathcal{I}}}}}}}}\)-symmetric crystal, where the global \({{{{{{{\mathcal{I}}}}}}}}\) center is represented with a red × symbol. d–h Bulk parity (\({{{{{{{\mathcal{I}}}}}}}}\)) eigenvalues and periodic-boundary-condition (PBC) energy spectra for the defect in b when the bulk is equivalent to d a ∣C∣ = 1 Chern insulator with band inversion at Γ, f a ∣C∣ = 1 Chern insulator with band inversion at Y, g a weak y-directed array c of x-directed Su–Schrieffer–Heeger (SSH) chains⁶⁰. Anomalous 0D defect states h with charge ± e/2 are present in cases f, g, but not d, which instead exhibits the trivial PBC spectrum in e [Eq. (2)]. Specifically, the spectrum in e may be deformed to that of a trivial insulator (i.e. a finite-sized insulator without midgap 0D states or without an imbalance in the number of states above or below the gap) without breaking \({{{{{{{\mathcal{I}}}}}}}}\) symmetry or closing the bulk gap, whereas the spectrum in h cannot. Hence, as defined in refs. 17, 39, 47, the midgap dislocation states in h are filling-anomalous. Next, by considering the limit in c in which the bulk is equivalent to a decoupled array of SSH chains, we find that the two dislocations correspond to the ends of a "leftover'' SSH chain that is decoupled from the bulk. This implies that the ± e/2-charged defect states are equivalent to the end states of an \({{{{{{{\mathcal{I}}}}}}}}\)-symmetric SSH chain⁶⁰ (red line in b), and thus persist under the relaxation of particle-hole symmetry^17,34,39,59. The explicit details of the numerical calculations shown in this figure are provided in SN 4A1.