Fig. 6: 0D dislocation states in 3D SnTe crystals.

a Defect geometry for an \({{{{{{{\mathcal{I}}}}}}}}\)-related pair of internal edge dislocations with B = a₁ in 3D SnTe, where the \({{{{{{{\mathcal{I}}}}}}}}\) center is marked with a red × symbol. In a, the sites enclosed within the black line have been removed in a finite number of layers in the tight-binding calculation to implement the pair of edge dislocations. b The PBC dislocation spectrum of SnTe using the edge dislocation geometry in a exhibits four filling-anomalous states (two Kramers pairs), consistent with Eq. (6) [see SN 6B2 for calculation details]. c The real-space profile of the four anomalous states in b. In c, two total Kramers pairs of states are localized on \({{{{{{{\mathcal{I}}}}}}}}\)-related dislocation corners (one Kramers pair of states is bound to every other corner). When the HEND states in c are half-filled, each Kramers pair corresponds to a chargeless, spin-1/2 quasiparticle (i.e. a spinon) that is equivalent to the corner state of an \({{{{{{{\mathcal{I}}}}}}}}\)- and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetric 2D FTI (see SN 4B2 and refs. 17, 23). d The SnTe defect plane, for which a cross-sectional cut is enclosed by the black lines in a, schematically depicted as a stack of PbTe monolayer defect lines (Fig. 3c, e). In d, each defect line has two 0D dislocations on its end, which each bind first-order 0D topological dislocation states. We choose PbTe for the monolayers—rather than SnTe— because a decoupled stack of PbTe monolayers has the same x, y components of the \({{{{{{{{\boldsymbol{M}}}}}}}}}_{\nu }^{{{{{{{{\rm{F}}}}}}}}}\) vector as a tetragonal supercell of 3D SnTe, whereas the interlayer coupling in realistic 3D PbTe drives additional band inversions [Eqs. (5) and (9), see Fig. 3 and SN 6 for further details]. Hence, HEND dislocation states can be considered the result of stacking and symmetrically coupling (gray arrows in d) an odd number of 2D monolayers that each contain first-order dislocation bound states.