Topological edge states of massless fermion with non-quantized and zero Berry phases

We investigate the bulk-boundary correspondence for massless Dirac fermion in $\alpha$-$T_3$ lattice where the Berry phase can be continuously tuned from $\pi$ (graphene) to $0$ ($T_3$ or dice lattice) without modifying the energy dispersion. The topological origin of edge states is revealed by unitary transform of the Hamiltonian, which maps $\alpha$-$T_3$ zigzag ribbon (ZR) into stub Su-Schrieffer-Heeger (SSH) chain. In the Majorana representation of the eigenstates, the ${\mathbb{Z}}_2$ invariant is manifested by azimuthal winding numbers on the Bloch sphere. Particularly, $T_3$ ZR exemplifies a topologically nontrivial system with zero Berry phase. Contrary to the conventional topological insulators, band gap closing in the corresponding stub SSH chain is not accompanied by a topological phase transition.