Interaction-controlled localization in one-dimensional chain: From edges to domain walls

Using Hartree-Fock mean-field approach, we study the role of on-site ($U$) and extended ($V$) Hubbard interactions on the existence and evolution of edge modes in a half-filled Su-Schrieffer-Heeger (SSH) chain. We analyze the energy spectrum, local probability amplitudes, and site-resolved charge and spin density profiles across topological, critical, and trivial hopping regimes. We find that the localization of bound states is controlled by the ratio $2V/U$, with edge spin-density-wave modes for $U>2V$ and mid-chain charge-density-wave domain walls for $U<2V$, independent of band topology. These results establish the correlation-driven origin of localized states in finite one-dimensional chains.