Majorana representation for topological edge states in Dirac fermion with non-quantized Berry phase

Research on the Dirac fermions has expanded to encompass systems that are described by three-level Hamiltonians, including α-T₃ lattice realized in Hg_1−xCd_xTe at a critical doping x = 0.17 [Nat. Phys. 10, 233 (2014), PRB 92, 035118 (2015)]. The α-T₃ lattice is constructed by connecting an additional atom to one of two atoms in each unit cell of the honeycomb lattice, with relative hopping strength 0 ≤ α ≤ 1. By tuning α, the α-T₃ lattice interpolates graphene (pseudospin S=1/2) and dice lattice (S=1) for α=0 and α=1, respectively. This is followed by a transition from diamagnetic to paramagnetic orbital susceptibility as a manifestation of the continuous change of the Berry phase γ from π to 0 [PRL 112, 026402 (2014)]. Thus, it is interesting to study whether topologically protected edge states exist in α-T₃ ribbons, where γ is no longer quantized. However, the bulk-boundary correspondence is yet to be formulated.

Here, we investigate the bulk-boundary correspondence in zigzag ribbon of α-T₃ lattice. By imposing boundary conditions on the wavefunction, we analytically determine the ranges of edge states in the Brillouin zone. We find that the zigzag ribbon can be mapped into a variant of the Su-Schrieffer-Heeger chain [PR Research 4, 013185 (2022)] by unitary transforms of the Hamiltonian. Finally, we prove that the topologically trivial and non-trivial phases of the zigzag ribbon (indicated by the absence and presence of the edge states, respectively), can be distinguished from the Majorana representation for the eigenstate [PRB 104, 195131 (2021)], where the Z₂ invariant is defined from the azimuthal winding number of the eigenstate on the Bloch sphere.