Studying the Su-Schrieffer-Heeger chain I have learned that the model has two different phases, one which is called topological and the other one trivial. In the notes it says that these phases are very different, and indeed they have different values for a topological invariant.
The phases are also physically different, in which the topological phase has a degenerate ground state and the trivial one does not. However, I don't understand what are so exciting about this degenerate ground state and why we care that much about it. I would assume that this could give room for zero-energy excitations, which may give rise to some current, but I don't see how this would be relevant for an edge of a one-dimensional system.