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

TL;DR

This paper explores the topological properties of massless Dirac fermions in the $ ext{α-}T_3$ lattice, revealing how edge states relate to Berry phase tuning without energy gap closure, challenging traditional topological insulator concepts.

Contribution

It demonstrates a topologically nontrivial phase with zero Berry phase in the $ ext{α-}T_3$ lattice, linking bulk-boundary correspondence to a stub SSH chain via a unitary transformation.

Findings

Edge states persist despite zero Berry phase.

The $ ext{α-}T_3$ lattice maps to a stub SSH chain.

Band gap closing does not imply a topological phase transition.

Abstract

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.