Abstract

We study the bulk-boundary correspondences for zigzag ribbons (ZRs) of massless Dirac fermion in two-dimensional α-T₃ lattice. By tuning the hopping parameter α ∈ [0, 1], the α-T₃ lattice interpolates between pseudospin S = 1/2 (graphene) and S = 1 (T₃ or dice lattice), for α = 0 and 1, respectively, which is followed by 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 not in {0, 1}. The α-T₃ ZRs are mapped into stub Su-Schrieffer-Heeger chains by performing unitary transforms of the bulk Hamiltonian. The non-trivial topology of the bulk bands is revealed by the Majorana representation of the eigenstates, where the Z₂ topological invariant is manifested by the winding numbers on the complex plane and the Bloch sphere.

Abstract

We study the bulk-boundary correspondences for zigzag ribbons (ZRs) of massless Dirac fermion in two-dimensional α-T₃ lattice. By tuning the hopping parameter α ∈ [0, 1], the α-T₃ lattice interpolates between pseudospin S = 1/2 (graphene) and S = 1 (T₃ or dice lattice), for α = 0 and 1, respectively, which is followed by 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 not in {0, 1}. The α-T₃ ZRs are mapped into stub Su-Schrieffer-Heeger chains by performing unitary transforms of the bulk Hamiltonian. The non-trivial topology of the bulk bands is revealed by the Majorana representation of the eigenstates, where the Z₂ topological invariant is manifested by the winding numbers on the complex plane and the Bloch sphere.

Abstract

We study the bulk-boundary correspondences for zigzag ribbons (ZRs) of massless Dirac fermion in two-dimensional α-T₃ lattice. By tuning the hopping parameter α ∈ [0, 1], the α-T₃ lattice interpolates between pseudospin S = 1/2 (graphene) and S = 1 (T₃ or dice lattice), for α = 0 and 1, respectively, which is followed by 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 not in {0, 1}. The α-T₃ ZRs are mapped into stub Su-Schrieffer-Heeger chains by performing unitary transforms of the bulk Hamiltonian. The non-trivial topology of the bulk bands is revealed by the Majorana representation of the eigenstates, where the Z₂ topological invariant is manifested by the winding numbers on the complex plane and the Bloch sphere.

Abstract

We study the bulk-boundary correspondences for zigzag ribbons (ZRs) of massless Dirac fermion in two-dimensional α-T₃ lattice. By tuning the hopping parameter α ∈ [0, 1], the α-T₃ lattice interpolates between pseudospin S = 1/2 (graphene) and S = 1 (T₃ or dice lattice), for α = 0 and 1, respectively, which is followed by 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 not in {0, 1}. The α-T₃ ZRs are mapped into stub Su-Schrieffer-Heeger chains by performing unitary transforms of the bulk Hamiltonian. The non-trivial topology of the bulk bands is revealed by the Majorana representation of the eigenstates, where the Z₂ topological invariant is manifested by the winding numbers on the complex plane and the Bloch sphere.