Are edge states always topological? Hey am new to this forum but I have a question regarding topologically protected states.. Lets suppose we have a 1D gapped system divided two to distinct regions that have different periodicity or different properties and that at the centre, where the two regions 'meet' states appear in the gap. The simplest example where this happens is the SSH model. But i was wondering whether this is generally true. Are these states always topological?
 A: No.
A simple example is given by the spin-$2$ Affleck-Kennedy-Lieb-Tasaki state/chain. For a while people thought this spin chain was topological, since it indeed has has degenerate edge modes with open boundary conditions, but we now know they are not protected and indeed there is no topological invariant associated to the system. More concretely, in ''Symmetry protection of topological order in one-dimensional quantum spin systems'' (Pollmann, Berg, Turner, Oshikawa, 2009) it was shown (in Fig 4) that one can adiabatically connect that spin chain to a trivial state, even without breaking spin rotation symmetry or any other symmetry that one might deem relevant.
Hence degenerate edge modes are not enough to label a chain as topological. A more secure way is by looking its entanglement spectrum when partitioning the chain in two halves. A topological chain will have degenerate entanglement values everywhere in the entanglement spectrum. I do not know of a single counter-example where a system has that property but is not topological. The above spin-$2$ chain, however, does have degenerate values at the bottom of the entanglement spectrum, but not higher up, consistent with it not being a (symmetry protected) topological phase of matter.
