  Studying the edge states of a topological system and extracting their
topological properties is of great importance in understanding and
characterizing these systems. In this paper, we present a novel analytical
approach for obtaining explicit expressions for the edge states in the
Kane-Mele model within a ribbon geometry featuring armchair boundaries. Our
approach involves a mapping procedure that transforms the system into an
extended Su-Schrieffer-Heeger model, specifically a two-leg ladder, in momentum
space. Through rigorous derivation, we determine various analytical properties
of the edge states, including their wave functions and energy dispersion.
Additionally, we investigate the condition for topological transition by solely
analyzing the edge states, and we elucidate the underlying reasons for the
violation of the bulk-edge correspondence in relatively narrow ribbons. Our
findings shed light on the unique characteristics of the edge states in the
quantum spin Hall phase of the Kane-Mele model and provide valuable insights
into the topological properties of such systems.
