Emerging topological bound states in Haldane model zigzag nanoribbons

The discovery of topological insulators and superconductors revolutionized our understanding of matter, highlighting the limitations of the Ginzburg-Landau classification. Since the first proposal for a topological insulator, significant breakthroughs have been made, including the classification of topological phases by symmetry classes, the discovery of higher-order topology, and the exploration of non-Hermitian topology. Dimensionality plays a critical role in defining topological invariants, typically defined for systems compactified in all directions. The bulk-boundary correspondence dictates that non-trivial, semi-infinite systems exhibit metallic boundary states. These states persist unless the uncompactified dimension becomes comparable to their decay length. Recent advancements in nanostructuring topological materials allow the creation of samples where this condition is not met, leading to active research on finite-size effects in topological phases. Dimensional crossovers between topological phases offer a promising avenue for engineering novel systems. Extensive studies on graphene nanoribbons (GNRs) have revealed a rich phenomenology, including robust topological bound states dependent on ribbon width and termination, detectable with local probes. Even before the concept of symmetry-protected topological insulators, the quantum Hall system and, notably, the Haldane model (1988), were studied. The Haldane model, proposing a quantum anomalous Hall phase, is a prominent example of a Chern insulator. Its time-reversal doubling yields the Kane-Mele model, a prototypical model for time-reversal-protected topological phases. While originally proposed for graphene (where the predicted spin Hall phase is unobservable due to weak spin-orbit coupling), it has recently found experimental realization in bismuthene and germanene. Haldane model nanoribbons provide a significant model for studying honeycomb-based topological materials and offer a theoretical platform to merge the physics of nanostructured topological insulators and GNRs. This paper focuses on zigzag Haldane nanoribbons, investigating the effects of dimensional reduction on the Haldane model's topological phase.

Several studies have examined finite-size effects on topological phases, including the impact on helical edge states in quantum spin-hall systems, dimensional crossover in topological insulator nanofilms, and the connection between higher-order topological insulators and lower-dimensional counterparts. Research has also explored interacting topological edge channels, the lifting of topological protection by edge coupling, and the relaxation and revival of quasiparticles in interacting quantum Hall liquids. Specific investigations have focused on Majorana modes in quantum spin Hall trenches, the role of edges in quasicrystalline Haldane models, and interference patterns in helical Josephson junctions. The tunability of bound states in second-order topological insulators and superconductors and the presence of multiple higher-order topological phases have also been studied. Finally, the observation of dimension crossover in a tunable 1D Dirac fermion and topological phase transitions in chiral graphene nanoribbons are relevant to this work. These previous works laid the groundwork for understanding the complexities of finite-size effects and dimensional crossovers in topological systems, providing a context for the current investigation.

The study employs the Haldane model, describing spinless fermions on a honeycomb lattice with an orthogonal periodic magnetic field and staggered on-site potential. The Hamiltonian includes nearest-neighbor and next-nearest-neighbor hoppings, parameterized by t₁ and t₂, respectively, and a phase φ accounting for staggered magnetic flux. The staggered on-site potential, m, breaks inversion symmetry. The topological phase, occurring when |m|/t₂ < 3√3 sin φ, exhibits a gapped bulk and non-trivial Chern number (c = ±1). A strip geometry with periodic boundary conditions (PBC) along the a₁ direction is considered. The Fourier transformation converts the real-space Hamiltonian to a Bloch Hamiltonian in k-space. Numerical tight-binding diagonalization is used to obtain energy bands. The localization length of the edge states, ξloc, is estimated using an existing formula. To analyze the topological properties of the zigzag Haldane strips, the energy gap Δ and the Zak phase φ are computed numerically. The Zak phase, a topological invariant for 1D systems, is calculated using a multi-band approach, considering a discretization of the Brillouin zone. The periodic gauge is enforced to define the Zak phase, which is 0 or π due to the Hamiltonian's real nature. The large-m limit is used to identify trivial and non-trivial phases based on the Zak phase difference. Numerical diagonalization of the model with open boundary conditions (OBC) along the a₁ direction is performed to investigate the presence of bound states. The robustness of these states against on-site disorder is tested numerically. A low-energy effective model, considering the coupling between two 1D chains representing the edges, is developed to qualitatively interpret the results. This model incorporates an exponentially decaying coupling with a sharp cutoff based on the minimum number of first-neighbor hoppings. Finally, the presence of Jackiw-Rebbi bound states at domain walls in the on-site staggered potential is investigated.

Numerical calculations reveal a width-dependent reentrant topological phase diagram. The number of gap closings and reopenings increases with strip width. Strips with N_y = 4M + 2 are gapless at m = 0. The Zak phase exhibits π jumps at each gap closing, indicating topological phase transitions. Based on Zak phase analysis, a phase diagram is constructed showing regions of topologically non-trivial phases. In these regions, open boundary condition (OBC) calculations show the emergence of quasi zero-dimensional (0D) bound states, localized at the strip ends, forming degenerate doublets. These states are robust against on-site disorder. The observed phenomenology is unique to zigzag nanoribbons, not observed in armchair nanoribbons. A low-energy effective model, considering the coupling between chiral edge states, qualitatively captures the behavior of the effective mass term, including the number of nodes, the spreading in k-space with increasing t₂, and the condition M(0; π) = 0 for N_y = 4M + 2. The model explains the mass inversion leading to the reentrant topological phase diagram. The effective model explains the asymmetry between zigzag and armchair nanoribbons, linking it to differences in localization length and the position of the Dirac point. Furthermore, bound states are observed at domain walls in the on-site staggered potential, consistent with the Jackiw-Rebbi mechanism, demonstrating fractional charge. These states persist even when the mass interpolates between regions lacking bound states in the uniform case.

The findings reveal a complex interplay between finite-size effects and topological properties in the Haldane model. The width-dependent reentrant topological phase diagram and the emergence of robust 0D bound states challenge simplistic low-energy theories and highlight the rich physics of coupled topological edges. The robustness of these states against disorder suggests potential applications in topological quantum computing. The observation of Jackiw-Rebbi-like bound states at domain walls further underscores the topological nature of these in-gap states. The asymmetry between zigzag and armchair nanoribbons underscores the importance of edge geometry in determining the topological properties of these systems. This research could significantly impact transport properties in constricted systems and studies on Majorana zero modes and parafermions.

This study demonstrates a reentrant topological phase diagram in zigzag Haldane nanoribbons driven by finite-size effects. The emergence of robust, quasi-0D bound states, their robustness to disorder, and the presence of Jackiw-Rebbi-like states at domain walls are key contributions. The effective model provides a qualitative understanding of the observed phenomena. Future research could explore the influence of interactions, explore different lattice geometries, and investigate potential applications in topological quantum devices.

The effective low-energy model provides a qualitative description but does not fully capture the quantitative details. The study focuses primarily on numerical simulations, with limited analytical solutions. The exploration of armchair nanoribbons was less extensive than that of zigzag nanoribbons. Further research is needed to fully understand the interplay between the various model parameters and their impact on the observed phenomena.