Emergent $\mathbb{Z}$-type topology in a quasi-one-dimensional extended QWZ model
We investigate the emergence of zero-dimensional topological end states in nanoribbons described by the Qi-Wu-Zhang (QWZ) model and its extensions with longer-range couplings. While dimensional reduction from two to one dimension is often assumed to preserve the symmetry classification of the parent system, here an additional symmetry can emerge originating from the real-space geometry of the ribbon. This symmetry acts as a chiral symmetry, combining orbital and spatial transformations, and promotes the effective one-dimensional system from symmetry class D to class BDI. We demonstrate that the existence of such a symmetry depends both on the long and end termination of the ribbon and exhibits an even-odd effect with respect to ribbon width, revealing that the commonly studied rectangular ribbons constitute a special high-symmetry case. For the conventional QWZ model, we derive analytic expressions for the topological phase boundaries of finite-width nanoribbons and characterize the resulting hybridization-gap phases through ($\mathbb{Z}_2 $) and winding-number invariants. We further show that extended QWZ models with longer-range couplings support phases with multiple topological end states and higher winding numbers. These phases arise through distinct mechanisms, including the hybridization of multiple edge modes inherited from higher-Chern-number bulk phases. Our results demonstrate that both long and end termination can fundamentally alter the topological classification of confined Chern insulators, highlighting the interplay between crystalline geometry, emergent symmetries, and dimensional reduction.