Topological edge states of massless fermions with nonquantized and zero Berry phases

We investigate the bulk-boundary correspondence for massless Dirac fermions in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>α</a:mi><a:mtext>−</a:mtext><a:msub><a:mi>T</a:mi><a:mn>3</a:mn></a:msub></a:mrow></a:math> lattice where the Berry phase can be continuously tuned from <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>π</b:mi></b:math> (graphene) to 0 (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi>T</c:mi><c:mn>3</c:mn></c:msub></c:math> or dice lattice) without modifying the energy dispersion. The topological origin of edge states is revealed by the unitary transformation of the Hamiltonian, which maps an <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>α</d:mi><d:mtext>−</d:mtext><d:msub><d:mi>T</d:mi><d:mn>3</d:mn></d:msub></d:mrow></d:math> zigzag ribbon (ZR) onto a stub Su-Schrieffer-Heeger (SSH) chain. In the Majorana representation of the eigenstates, the topological invariant is manifested by the azimuthal winding number on the Bloch sphere. Particularly, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:msub><e:mi>T</e:mi><e:mn>3</e:mn></e:msub></e:math> ZR exemplifies a topologically nontrivial system with zero Berry phase. Contrary to conventional topological insulators, the gap closing in the corresponding stub SSH chain is not accompanied by a topological phase transition. Published by the American Physical Society 2024