Topological junctions for one-dimensional systems

We study and classify the emergence of protected edge modes at the junction of one-dimensional materials. Using symmetries of Lagrangian planes in boundary symplectic spaces, we present a novel proof of the periodic table of topological insulators in one dimension. We show that edge modes necessarily arise at the junction of two materials having different topological indices. Our approach provides a systematic framework for understanding symmetry-protected modes in one dimension. It does not rely on periodic nor ergodicity and covers a wide range of operators which includes both continuous and discrete models.