From Chern to Winding: Topological Invariant Correspondence in the Reduced Haldane Model
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| Format: | Preprint |
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2025
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| _version_ | 1866912395904942080 |
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| author | Al-Mahmood, Ghassan Amini, Mohsen Ghanbari-Adivi, Ebrahim Soltani, Morteza |
| author_facet | Al-Mahmood, Ghassan Amini, Mohsen Ghanbari-Adivi, Ebrahim Soltani, Morteza |
| contents | 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 ($ν$) 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 $ν$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_20472 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From Chern to Winding: Topological Invariant Correspondence in the Reduced Haldane Model Al-Mahmood, Ghassan Amini, Mohsen Ghanbari-Adivi, Ebrahim Soltani, Morteza Mesoscale and Nanoscale Physics Quantum Physics 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 ($ν$) 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 $ν$ 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. |
| title | From Chern to Winding: Topological Invariant Correspondence in the Reduced Haldane Model |
| topic | Mesoscale and Nanoscale Physics Quantum Physics |
| url | https://arxiv.org/abs/2505.20472 |