Exotic edge states of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math> high-fold fermions in honeycomb lattices

A generalization of the graphene honeycomb model to the case where each site in the honeycomb lattice contains a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>n</a:mi></a:math>-fold degenerate set of eigenstates of the <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msub><b:mi>C</b:mi><b:mn>3</b:mn></b:msub></b:math> symmetry has been recently proposed to describe several systems, including triangulene crystals and photonic lattices. These generalized honeycomb models are defined by <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mo>(</c:mo><c:msub><c:mi>n</c:mi><c:mi>a</c:mi></c:msub><c:mo>,</c:mo><c:msub><c:mi>n</c:mi><c:mi>b</c:mi></c:msub><c:mo>)</c:mo></c:mrow></c:math>, the number of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msub><d:mi>C</d:mi><d:mn>3</d:mn></d:msub></d:math> eigenstates in the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>a</e:mi></e:math> and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>b</f:mi></f:math> sites of the unit cell, resulting in <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:msub><g:mi>n</g:mi><g:mi>a</g:mi></g:msub><g:mo>+</g:mo><g:msub><g:mi>n</g:mi><g:mi>b</g:mi></g:msub></g:mrow></g:math> bands. Thus, the (1,1) case gives the coventional honeycomb model that describes the two low-energy bands in graphene. Generalizations, such as (2,1), (2,2), and (3,3) display several nontrivial features, such as coexisting graphenelike Dirac cones with flat bands, both at zero and finite energy, as well as robust degeneracy points where a flat band and a parabolic band meet at the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi mathvariant="normal">Γ</h:mi></h:math> point. Here we explore the edge states of this class of crystals, using as reference triangulene crystals, and we find several types of edge states absent in the conventional (1,1) honeycomb case, associated to the nontrivial features of the two-dimensional bands of the high-fold case. First, we find dispersive edge states associated to the finite-energy flat bands, that occur both at the armchair and zigzag termination. Second, in the case of noncentrosymmetric triangulene crystals that lead to a <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mi>S</j:mi><j:mo>=</j:mo><j:mn>1</j:mn></j:mrow></j:math> Dirac band, we have a bonding-antibonding pair of dispersive edge states, localized in the same edge so that their energy splitting is reduced as their localization increases, opposite to the conventional behavior of pairs of states localized in opposite edges. Third, for the (3,3) case, that hosts a gap separating a pair of flat conduction and valence bands, we find nondispersive edge states with <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mi>E</k:mi><k:mo>=</k:mo><k:mn>0</k:mn></k:mrow></k:math> in all edge terminations. Published by the American Physical Society 2024