Numerical Investigation of Topological Phase Transitions, Disorder Robustness, and Bulk-Boundary Correspondence in the One-Dimensional Su-Schrieffer-Heeger Model

This research paper presents a numerical investigation of the one-dimensional Su-Schrieffer-Heeger (SSH) model using exact matrix diagonalization in Python. By simulating a 1D dimerized atomic chain under open boundary conditions (N=20 unit cells), the study explores the emergence of topologically protected zero-energy (E \approx 0) edge states in the topological regime (v < w) and demonstrates their spatial localization at the chain boundaries. To test the resilience of these edge modes, random off-diagonal disorder is injected into the Hamiltonian, proving that chiral symmetry prevents the mid-gap states from shifting away from zero energy despite heavy structural noise. Finally, by transitioning to momentum space (k-space) under periodic boundary conditions, the paper maps the 2D pseudospin vector field d(k) = (d_x(k), d_y(k)) across the Brillouin zone, establishing the Bulk-Boundary Correspondence through the non-trivial winding number (W = 1, Zak phase \mathcal{Z} = \pi).