From Chern to Winding: Topological Invariant Correspondence in the Reduced Haldane Model

• URL: http://arxiv.org/abs/2505.20472v1
• Date: Mon, 26 May 2025 19:11:43 GMT
• Title: From Chern to Winding: Topological Invariant Correspondence in the Reduced Haldane Model
• Authors: Ghassan Al-Mahmood, Mohsen Amini, Ebrahim Ghanbari-Adivi, Morteza Soltani,
• Abstract summary: We present an exact analytical investigation of the topological properties and edge states of the Haldane model defined on a honeycomb lattice with zigzag edges.<n>We show that the $nu$ exactly reproduces the Chern number of the parent model in the topologically nontrivial phase.<n>Our analysis further reveals the critical momentum $ k_c $ where edge states traverse the bulk energy gap.
• Score: 0.4249842620609682
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present an exact analytical investigation of the topological properties and edge states of the Haldane model defined on a honeycomb lattice with zigzag edges. By exploiting translational symmetry along the ribbon direction, we perform a dimensional reduction that maps the two-dimensional model into a family of effective one-dimensional systems parametrized by the crystal momentum $k_x$. Each resulting one-dimensional Hamiltonian corresponds to an extended Su-Schrieffer-Heeger (SSH) model with momentum-dependent hoppings and onsite potentials. We introduce a natural rotated basis in which the Hamiltonian becomes planar and the winding number ($\nu$) is directly computable, providing a clear topological characterization of the reduced model. This framework enables us to derive closed-form expressions for the edge-state wavefunctions and their dispersion relations across the full Brillouin zone. We show that the $\nu$ exactly reproduces the Chern number of the parent model in the topologically nontrivial phase and allows for an exact characterization of the edge modes. Analytical expressions for the edge-state wavefunctions and their dispersion relations are derived without requiring perturbative methods. Our analysis further reveals the critical momentum $ k_c $ where edge states traverse the bulk energy gap, and establishes precise conditions for the topological phase transition. In contrast to earlier models, such as plaquette-based tight-binding reductions, our method reveals hidden geometric symmetries in the extended SSH structure that are essential for understanding the topological behavior of systems with long-range hopping. Our findings offer new insight into the topological features of zigzag nanoribbons and establish a robust framework for analyzing analogous systems.

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