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

TL;DR

This paper provides an exact analytical framework to connect the topological invariants of the Haldane model with its edge states by reducing it to a family of extended SSH models, enabling precise characterization of topological phases.

Contribution

It introduces a novel analytical approach that maps the 2D Haldane model to 1D effective models, allowing exact calculation of topological invariants and edge states without perturbation.

Findings

Exact expressions for edge-state wavefunctions and dispersion relations.

The winding number $ u$ matches the Chern number in the topological phase.

Identification of critical momentum $k_c$ for edge state traversal.

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.