Emerging topological bound states in Haldane model zigzag nanoribbons

The work investigates how finite-size effects in zigzag nanoribbons implementing the Haldane (Chern insulator) model reshape topological edge physics. While in wide systems chiral edge modes traverse the bulk gap, reducing the width can induce hybridization between counter-propagating edge states, potentially opening an edge gap. The research question is whether such finite-size-induced gaps are topologically trivial or non-trivial, how this depends on parameters (notably the staggered sublattice potential m), and how to characterize the resulting phases. The study is motivated by advances in topological materials and nanoribbon physics, including graphene nanoribbons and honeycomb-lattice topological insulators such as those described by Haldane and Kane–Mele models. The authors find that thin zigzag Haldane nanoribbons exhibit a reentrant, width-dependent topological phase diagram as a function of m, with topological transitions marked by π jumps in the Zak phase and the emergence (or absence) of quasi-0D end states within the finite-size-induced edge gap. Domain-wall configurations of m yield Jackiw–Rebbi bound states. The results highlight the crucial role of system width and edge termination in governing low-dimensional topological phenomena.

The introduction reviews key developments: discovery and classification of topological insulators and superconductors; higher-order and non-Hermitian topology; the bulk-boundary correspondence and finite-size effects on topological phases. It recalls the Haldane model as a seminal Chern insulator and its time-reversal doubled Kane–Mele model, noting experimental realizations (e.g., bismuthene, germanene). Prior studies on graphene nanoribbons showed rich width- and termination-dependent topological features, including robust bound states detectable by local probes. Finite-size effects have been widely explored in topological systems, revealing dimensional crossovers and edge coupling phenomena. This context motivates studying Haldane model zigzag nanoribbons as an ideal platform to merge Chern-insulator physics with nanoribbon phenomenology.

- Model: Spinless Haldane model on a honeycomb lattice with nearest-neighbor hopping t1, next-nearest-neighbor hopping t2 with phase φ, and staggered on-site potential m breaking inversion symmetry. The 2D topological phase occurs for |m|/t2 < 3√3 sin φ (Chern number c = ±1). Zigzag nanoribbons are considered with periodic boundary conditions (PBC) along a1 (x) and finite width Ny along a2 (y); later, open boundary conditions (OBC) along a1 are used to probe end states. - Numerical diagonalization: Tight-binding Hamiltonians for strips are constructed and diagonalized (PBC and OBC). Parameters typically fixed to t1 = 1 (energy unit), t2 = 0.3, φ = π/2 to maximize the Haldane bulk gap; widths Ny = 4, 6, 8, 10 are examined. Under PBC along x, band structures are computed; under OBC along x, low-energy spectra are obtained for finite lengths (e.g., 20a, 40a, 80a, 160a) to search for in-gap end states. - Gap and phase characterization: The edge-induced gap Δ is computed versus m for each Ny. The Zak phase φZak for the occupied bands is used as a 1D topological indicator. For a multiband system, a discretized Wilson-loop approach is employed: define k-grid k_j = 2πj/N, compute occupied-band eigenvectors |u_{n,k_j}⟩, enforce periodic gauge |u_{n,k_j}⟩ = e^{−ik_j x} |ũ_{n,k_j}⟩ (x_e being the site positions in the unit cell), build overlap matrices S(k_j, k_{j+1})_{mn} = ⟨u_{m,k_j}|u_{n,k_{j+1}}⟩, then φZak = −i log det[∏_j S(k_j, k_{j+1})]. For the real Hamiltonian considered, φZak ∈ {0, π}. Zak phase changes are referenced to the large-|m| trivial limit, with care for Ny = 4M+2 where m → ±∞ give different reference values. - End states under OBC: For m values within regions identified as topological by φZak, OBC spectra are computed. In-gap eigenvalue doublets are identified and their probability densities are mapped along x to confirm exponential localization at strip ends. - Robustness to disorder: Random on-site disorder is added to test stability of in-gap states; bound states persist unless disorder strength approaches the gap size (details in Supplementary Note 3). - Low-energy effective theory: An effective model for coupled zigzag edge modes is formulated. A minimal Hamiltonian near the Dirac point includes an effective hybridization mass M(k) induced by edge coupling. A phenomenological construction models two edge chains coupled with exponentially decaying, range-cutoff hoppings, leading to analytic forms for M(k) that depend on Ny class (Ny = 4M vs Ny = 4M+2). Numerical extraction of |M(m; k)| is performed from the lowest positive band E_Ny(m; k) via |M| = sqrt[E_Ny(m; k)^2 − E_∞(m; k)^2], and compared to analytic tendencies (node count, k-spread, M(0; π) = 0 for Ny = 4M+2). - Domain-wall (Jackiw–Rebbi) setup: A spatially varying m(x) is implemented, interpolating between regions with different Zak phases (e.g., m = −0.3 on left, m = +0.3 on right, with smoothing at interface) for Ny = 10 and length L = 200a. Real-space diagonalization locates in-gap bound states localized at the domain wall. - Tools: Pybinding used for finite-size model construction and diagonalization; Zak phase computations validated with original code and benchmarks. Data and codes available on request.

- Reentrant width-dependent topology: As a function of the staggered potential m, thin zigzag Haldane nanoribbons exhibit multiple gap closings and reopenings whose number increases with strip width (Ny = 4, 6, 8, 10). At each gap closing, the Zak phase jumps by π, indicating topological phase transitions and a reentrant phase diagram. - Special even-width behavior: For Ny = 4M+2 (e.g., Ny = 6, 10), the edge spectrum is gapless at m = 0 despite strong proximity of edge modes, due to symmetry-induced nonhybridization (proven analytically in Supplementary Note 1). For these widths, the large-|m| reference for the Zak phase differs between m → ±∞ (Supplementary Note 2). - Quasi-0D end states: In topologically non-trivial regions (identified by Zak phase relative to large-|m| trivial reference), OBC spectra display a degenerate doublet of in-gap eigenstates. Their probability densities are exponentially localized at the two strip ends, confirming quasi-0D bound states induced by finite-size edge hybridization. Examples shown for Ny = 4 (m ≈ 0), Ny = 6 (m ≈ 0.5), Ny = 8 (m ≈ 0.8), Ny = 10 (m ≈ 0.92) at t1 = 1, t2 = 0.3, φ = π/2. - Robustness to disorder: These end states persist under random on-site disorder; while energies shift slightly, the states remain unless disorder strength becomes comparable to the edge gap. - Low-energy mechanism: A phenomenological edge-coupling model captures key qualitative features of the effective hybridization mass M(k), including node counts versus width, spread in k with increasing t2, and M(0; π) = 0 for Ny = 4M+2. Numerics show that increasing m shifts the Dirac point and the mass profile oppositely in k-space, causing mass inversion at certain m and explaining reentrant topology. - Domain-wall bound states: A spatial mass inversion between regions with Zak phases differing by π yields an in-gap bound state localized at the interface (Jackiw–Rebbi mechanism), carrying fractional charge ±(e/2). Demonstrated for Ny = 10, L = 200a with m switching between −0.3 and +0.3. - Geometry sensitivity: The phenomena are highly sensitive to strip width and edge termination. Armchair nanoribbons do not display analogous reentrant behavior in the explored parameter range, attributed to distinct Dirac point behavior and localization-length trends.

The study addresses how finite-size-induced edge hybridization in zigzag Haldane nanoribbons can generate both topologically trivial and non-trivial gaps. By computing the Zak phase across m and correlating π jumps with gap closings, the authors establish a clear criterion for topological versus trivial gapped phases in the effectively 1D geometry. The emergence of exponentially localized end doublets under OBC in topological regions confirms the bulk-boundary correspondence in the reduced dimension. The effective low-energy framework explains the reentrant phase diagram via mass inversion: the interplay between the Dirac point shift (with m) and the momentum-dependent hybridization mass M(k) creates parameter windows where the mass changes sign at the Dirac point, toggling the topology. The absence of similar behavior in armchair ribbons underscores the role of edge termination and the specific dependence of edge-state localization length on t2. The domain-wall setup further validates the topological interpretation: when regions with Zak phases differing by π are interfaced, Jackiw–Rebbi bound states appear at the domain wall. These findings suggest that in nanostructured Chern insulators, edge coupling can yield rich, tunable low-dimensional topological physics beyond naive expectations, with implications for transport resonances in constricted devices and potential connections to superconducting platforms hosting Majorana or parafermionic modes.

The paper demonstrates that zigzag Haldane nanoribbons, upon dimensional reduction, exhibit a width-dependent reentrant topological phase diagram versus staggered potential m. Gap closings coincide with π jumps of the Zak phase and delineate alternating trivial and non-trivial phases. In the latter, quasi-0D, exponentially localized end-state doublets emerge within the finite-size-induced edge gap and are robust against on-site disorder. A phenomenological low-energy model captures the qualitative mechanism—mass inversion driven by the relative shift of the Dirac point and a momentum-dependent hybridization mass—and explains special behaviors for Ny = 4M+2. Domain walls between regions with differing Zak phases host Jackiw–Rebbi bound states. These results highlight strong sensitivity to strip width and edge termination, predicting significant consequences for transport in confined Chern-insulator devices and suggesting routes relevant to engineered topological superconductivity and non-Abelian excitations. Future work could explore broader parameter regimes (including armchair terminations), interactions, experimental realizations, and coupling to superconductivity to harness these bound states for quantum devices.

- The effective low-energy theory is qualitative: the hybridization mass M depends nontrivially on parameters and momentum, hindering a fully predictive analytical model; comparisons are mainly qualitative. - Results focus on specific parameter choices (e.g., t1 = 1, t2 = 0.3, φ = π/2) and a limited set of widths; generalization to broader ranges requires further study. - Armchair ribbons were only explored numerically in a limited parameter window and showed no analogous behavior there; a comprehensive analytical understanding for armchair geometry is not provided. - Disorder robustness was tested only for random on-site potentials; other disorder types (e.g., hopping disorder, edge roughness) and stronger disorder regimes are not systematically analyzed. - Spinless Haldane model idealization neglects interactions and spin; experimental platforms may introduce additional complexities. - The fractional charge assignment is inferred from Jackiw–Rebbi arguments; direct charge quantification beyond idealized models is not presented in the main text.