Fig. 3: First-order dislocation states in 2D PbTe monolayers.
From: Topological zero-dimensional defect and flux states in three-dimensional insulators
a The crystal structure of monolayer PbTe and the bulk BZ. The yellow diamond in a indicate the primitive cell. A PbTe monolayer40,41 has fourfold rotation, \({{{{{{{\mathcal{I}}}}}}}}\), and mirror symmetries (layer group \(p4/mmm1^{\prime}\)39). b Band structure of a PbTe monolayer along the high-symmetry lines of the 2D BZ in a. The size of the circle at each plotted point in b indicates the spectral weight on the Te (left panel) and Pb (right panel) atoms, and the color bars indicate the orbital character of each Bloch state on a scale of pz or px,y orbitals. We have additionally labeled the irreducible small corepresentations at the symmetry-independent TRIM points in the BZ in a (see SN 6A for details). The bands in b are inverted at X and \(X^{\prime}\), driving the bulk into a 2D mirror TCI phase40,41 with mirror Chern number \({C}_{{M}_{z}}=2\) and nontrivial weak (partial) SSH indices \({{{{{{{{\boldsymbol{M}}}}}}}}}_{\nu }^{{{{{{{{\rm{SSH}}}}}}}}}=({{{{{{{{\boldsymbol{b}}}}}}}}}_{1}+{{{{{{{{\boldsymbol{b}}}}}}}}}_{2})/2\) (see SN 3A and 6A). c Schematic of our real-space implementation of an \({{{{{{{\mathcal{I}}}}}}}}\)-related pair of B = a1 point dislocations in a Wannier-based tight-binding model of a PbTe monolayer obtained from first-principles calculations (details provided in SN 6A), where the \({{{{{{{\mathcal{I}}}}}}}}\) center is marked with a black × symbol. In c, the sites enclosed within the black line have been removed to implement the pair of point dislocations. d PBC energy spectrum of a tight-binding model of PbTe with the \({{{{{{{\mathcal{I}}}}}}}}\)-related pair of B = a1 point dislocations shown in c; there are four midgap, filling-anomalous17,23,39,47 dislocation states, consistent with Eq. (2) [\({{{{{{{\boldsymbol{B}}}}}}}}\cdot {{{{{{{{\boldsymbol{M}}}}}}}}}_{\nu }^{{{{{{{{\rm{SSH}}}}}}}}}\,{{{{{{{\rm{mod}}}}}}}}\ 2\pi=\pi\)]. e The real-space localization of the four midgap states in d, which subdivide into two \({{{{{{{\mathcal{I}}}}}}}}\)-related Kramers pairs. One Kramers pair of states is localized on each dislocation core and corresponds when half-filled to a chargeless, spin-1/2 quasiparticle (i.e. a spinon) that is equivalent to the end state of a spinful SSH chain.
