Abstract

We investigate the bulk-boundary correspondence for massless Dirac fermion in the α-T₃ lattice, where the Berry phase can be continuously tuned from π (graphene) to 0 (T₃ or dice lattice) without modifying the energy dispersion. By a unitary transformation, the α-T₃ zigzag ribbon is mapped to a stub SSH chain. In the Majorana representation of the eigenstates, the Z₂ invariant is manifested by azimuthal winding numbers on the Bloch sphere. The work shows that the T₃ zigzag ribbon is topologically nontrivial even with zero Berry phase, and that band-gap closing in the stub SSH chain is not accompanied by a topological phase transition.

Abstract

We investigate the bulk-boundary correspondence for massless Dirac fermion in the α-T₃ lattice, where the Berry phase can be continuously tuned from π (graphene) to 0 (T₃ or dice lattice) without modifying the energy dispersion. By a unitary transformation, the α-T₃ zigzag ribbon is mapped to a stub SSH chain. In the Majorana representation of the eigenstates, the Z₂ invariant is manifested by azimuthal winding numbers on the Bloch sphere. The work shows that the T₃ zigzag ribbon is topologically nontrivial even with zero Berry phase, and that band-gap closing in the stub SSH chain is not accompanied by a topological phase transition.

Abstract

We investigate the bulk-boundary correspondence for massless Dirac fermion in the α-T₃ lattice, where the Berry phase can be continuously tuned from π (graphene) to 0 (T₃ or dice lattice) without modifying the energy dispersion. By a unitary transformation, the α-T₃ zigzag ribbon is mapped to a stub SSH chain. In the Majorana representation of the eigenstates, the Z₂ invariant is manifested by azimuthal winding numbers on the Bloch sphere. The work shows that the T₃ zigzag ribbon is topologically nontrivial even with zero Berry phase, and that band-gap closing in the stub SSH chain is not accompanied by a topological phase transition.

Abstract

We investigate the bulk-boundary correspondence for massless Dirac fermion in the α-T₃ lattice, where the Berry phase can be continuously tuned from π (graphene) to 0 (T₃ or dice lattice) without modifying the energy dispersion. By a unitary transformation, the α-T₃ zigzag ribbon is mapped to a stub SSH chain. In the Majorana representation of the eigenstates, the Z₂ invariant is manifested by azimuthal winding numbers on the Bloch sphere. The work shows that the T₃ zigzag ribbon is topologically nontrivial even with zero Berry phase, and that band-gap closing in the stub SSH chain is not accompanied by a topological phase transition.