Electronic properties and topological aspects of graphene nanohelicoids

The gist The study introduces graphene nanohelicoids, geometric analogues of graphene nanoribbons, and investigates their electronic properties and topological aspects by constructing effective one-dimensional lattice models that exhibit anti-chiral symmetry.

Geometric Construction and Symmetry

Graphene nanohelicoids are defined as geometric analogues of graphene nanoribbons where the honeycomb lattice is embedded on a helicoidal surface. Starting from the three-dimensional helical structure, effective one-dimensional lattice models are constructed with a band structure characterized by a momentum-shifted particle–hole relation Ev(k) = −Ec(k + π) that reflects an anti-chiral symmetry arising from the nonsymmorphic symmetry. This anti-chiral symmetry is rooted in the anti-bipartite lattice structure of the effective model. The system exhibits a fractional translation symmetry combining a rotation in the xy-plane with a fractional translation along the z-axis, denoted as C6c6>.

Width Dependence and Electronic Regimes

The finite width of the helicoid serves as a direct geometric knob for tuning the low-energy spectrum, analogous to graphene nanoribbons. As the sample width varies, the system undergoes repeated transitions between semiconducting and metallic regimes through symmetry-constrained gap closings and openings. Specifically, when the number of atoms is even in the zigzag geometry, a finite gap near the Fermi level is exhibited, whereas when the number of atoms is odd, the band structure becomes gapless and a band necessarily crosses the Fermi level. The band dispersion evolves systematically with increasing helicoid width; as W increases, the effective dispersion along the helicoid axis becomes progressively flatter, leading to a step-like spectral profile in the large-width limit.

Edge Termination Effects

The electronic spectra differ qualitatively between zigzag and armchair edges because they impose different boundary conditions on the lattice. Zigzag edges preserve the valley degree of freedom, while armchair edges couple the two valleys K and K′ through the boundary condition. For zigzag edge GNHs, edge-localized states strongly affect the dispersion and lead to a pronounced width dependence. Conversely, armchair edges do not support robust edgelocalized zero-energy modes; instead, low-energy states originate from confined bulk Dirac states. The low-energy spectrum of armchair GNHs is controlled primarily by the discrete junction type rather than by the continuous increase of the helicoid width.

Topological Invariants and Zak Phase

The topological properties are calculated using a winding number formula derived from the Hamiltonian's symmetries. For even width W in the zigzag edge GNH, the winding number is given by ν = W/2. This results in a Zak phase being Z = π(W/2) (mod 2). The charge center is displaced by exactly half a lattice spacing of the effective 1D chain for W = 4m − 2, and localized at the unit cell for W = 4m. For the armchair-edge GNH, the winding number is given by ν = wII + 4/6.

Local Density of States Analysis

The local density of states (LDOS) analysis reveals that for the zigzag helicoid system, the LDOS associated with bands near the Fermi level is strongly concentrated along the zigzag edges. In contrast, for armchair-edge GNHs, the LDOS indicates that electronic states are predominantly distributed over the interior region of the lattice, reflecting bulk state features.

Conclusion

The work demonstrates that the electronic structure of finite width graphene helicoid can be understood from an effective one-dimensional lattice description that directly reflects the underlying geometry. The system exhibits an anti-chiral (or antibipartite) symmetry, which enforces a momentum-shifted spectral relation Ev(k) = −Ec(k + π). A key result is that the finite width of the helicoid acts as a geometric control parameter for the electronic spectrum. The related features follow directly from the lattice structure and its quantization condition, and can be understood analytically within the tight-binding description. This work highlights extended lattice defects as a viable platform for engineering tunable topological electronic states, opening new possibilities for geometry-driven design in graphene systems as a natural extension of the graphene nanoribbon geometry.

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Xiaoqian Liu,1,∗ Arsen Herasymchuk,2,∗ Yaroslav Zhumagulov,1 and Oleg V. Yazyev1,† Institute of Physics, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland 2 Department of Physics, University of Z¨urich, CH-8057 Z¨urich, Switzerland (Dated: July 16, 2026 We introduce graphene nanohelicoids, geometric analogues of graphene nanoribbons, in which the honeycomb lattice is embedded on a helicoidal surface. Starting from the three-dimensional helical structure, we construct effective one-dimensional lattice models with band structures characterized by a momentum-shifted particle–hole relation Ev(k) = −Ec(k + π) that reflects an anti-chiral symmetry arising from the nonsymmorphic symmetry. A systematic investigation of graphene nanohelicoids using the tight-binding approximation reveals a number of trends upon varying width and edge orientation, for instance, alternating transitions between semiconducting and metallic regimes. As the structure width varies, the band gap periodically closes and reopens, accompanied by an alternating Zak phase that switches between trivial and nontrivial. We derive an analytic tightbinding model and introduce a continuous deformation of the graphene nanohelicoids that explains the origin of width-dependent band inversion and alternating Zak phase. I INTRODUCTION Graphene nanoribbons have made a long way from starting as a highly popular theoretical model [1–4] to becoming a versatile experimental platform for controlling the electronic structure of graphene through geometry and boundary conditions [5–8]. Unlike infinite two-dimensional graphene, a graphene nanoribbon has a finite width and well-defined edges, so that its lowenergy spectrum is determined not only by the bulk Dirac bands, but also by boundary conditions and lattice termination [9–12]. Graphene nanoribbons with zigzag or low-symmetry edges support edge-localized states near the Fermi level, whereas armchair nanoribbons generally do not host such robust edge states. Their low-energy gaps are instead determined by how the finite ribbon width fits with the underlying graphene lattice [2, 3, 12].

1. Precision control over topological electronic states in graphene nanoribbons can be achieved by tuning their geometric width, which causes alternating transitions between semiconducting and metallic regimes as described in the paper. This allows for deterministic switching of bulk polarization through controlled manipulation of the system's physical size.

2. AI systems can be designed to predict the Zak phase evolution based on helicoid width or armchair structure index by utilizing the derived analytic tight-binding model and its momentum-shifted particle-hole relation Ev(k) = -Ec(k + pi). This enables the prediction of whether a system will exhibit a trivial or nontrivial bulk polarization.

3. Machine learning models can be trained to classify the topological invariant, such as the winding number nu, based on input parameters like helicoid width W or armchair index w. This allows for automated determination of boundary states localization patterns, specifically distinguishing between W/2 boundary states and type-II armchair-edge GNH localization atoms.

4. AI algorithms can simulate the local density of states (LDOS) for graphene nanohelicoids to map spatial electronic structure with high fidelity. The system can specifically distinguish between edge-localized modes governed by zigzag edges and bulk-like character originating from interior regions, guiding the focus of low-energy effective models.

5. AI systems can be used to generate optimized boundary conditions for graphene nanoribbons based on the discrete junction type (type I, II, or III). This capability allows for designing devices where specific junctions are engineered to be gapped or gapless, tailoring transport properties precisely to the required electronic outcome.

We introduce graphene nanohelicoids, geometric analogs of graphene nanoribbons, in which the honeycomb lattice is embedded on a helicoidal surface. Starting from the three-dimensional helical structure, we construct effective one-dimensional lattice models with band structures characterized by a momentum-shifted particle-hole relation E v(k)=-E c(k+π) that reflects an antichiral symmetry arising from the nonsymmorphic symmetry. A systematic investigation of graphene nanohelicoids using the tight-binding approximation reveals a number of trends upon varying width and edge orientation, for instance, alternating transitions between semiconducting and metallic regimes. As the structure width varies, the band gap periodically closes and reopens, accompanied by an alternating Zak phase that switches between trivial and nontrivial. Using an analytic tight-binding model with a continuous deformation of the graphene nanohelicoids, we explain the origin of width- and termination-dependent band inversion and alternating Zak phase, and derive the relationship between the termination-resolved index and the sublattice symmetry operator in one-dimensional nonsymmorphic systems.