Majorana representation for topological edge states of massless Dirac fermion with non-quantized Berry phase

TL;DR

This paper investigates the topological properties and edge states of the $ ext{alpha-}T_3$ lattice, revealing how the Berry phase and edge state transitions depend on the hopping parameter, using Majorana representation and bulk-boundary correspondence.

Contribution

It introduces a Majorana representation approach to analyze topological edge states in the $ ext{alpha-}T_3$ lattice, connecting bulk topology with edge state behavior across a range of parameters.

Findings

Edge state existence depends on the hopping parameter $ ext{alpha}$.

Transitions of in-gap bands occur at multiple points in momentum space.

Majorana representation reveals $ ext{Z}_2$ topological invariants through winding numbers.

Abstract

We study the bulk-boundary correspondences for zigzag ribbons (ZRs) of massless Dirac fermion in two-dimensional $\alpha$-${T}_3$ lattice. By tuning the hopping parameter $\alpha\in[0,1]$, the $\alpha$-${T}_3$ lattice interpolates between pseudospin $S=1/2$ (graphene) and $S=1$ (${T}_3$ or dice lattice), for $\alpha=0$ and $1$, respectively, which is followed by continuous change of the Berry phase from $\pi$ 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 $\alpha\notin \{0,1\}$. The $\alpha$-${T}_3$ 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 $\mathbb{Z}_2$ topological invariant is manifested by the winding numbers on the complex plane and the Bloch sphere.