Majorana representation for topological edge states of massless Dirac fermion with nonquantized Berry phase

We study the bulk-boundary correspondences for zigzag ribbons (ZRs) of massless Dirac fermion in the two-dimensional α − T 3 lattice. By tuning the hopping parameter α ∈ [ 0 , 1 ] , the α − T 3 lattice interpolates between pseudospin S = 1 / 2 (graphene) and S = 1 ( T 3 or dice lattice), for α = 0 and 1, respectively, which is followed by a continuous change of the Berry phase from π to 0. The range of existence for edge states in the momentum space is determined by solving tight-binding equations at the boundaries of the ZRs. We find that the transitions of in-gap bands from bulk to edge states in the momentum space do not only occur at the positions of the Dirac cones but also at additional points depending on α ∈ ( 0 , 1 ) . The α − T 3 ZRs are mapped onto stub Su-Schrieffer-Heeger chains by performing unitary transformations of the bulk Hamiltonian. The nontrivial topology of the bulk bands is revealed by the Majorana representation of the eigenstates, where the topological invariant is manifested by winding numbers on the complex plane and the Bloch sphere.