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I'm studying the topic of Topological insulators, I'm having a very hard time understanding what is the relationship between the fact that topological invariants are different from $0$ and the presence of edge states.

For instance in the Su Schrieffer model, all lecture notes and books I've read just limit themselves to calculate the topological invariants at different phases and then they just show that at this parameters it happens that there are edge states (example https://arxiv.org/abs/1509.02295).

This explanation to me just happens to show a 'coincidence', whenever there is a non zero topological invariant,there is presence of edge states. But then I don't understand the connection. Could someone clear up this for me? Or at least give links to literature that explain clearly this connection?

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To my knowledge there is no simple proof of the bulk-boundary correspondence. That's why it's so surprising and interesting. It is really weird that there is a correspondence because they are different setups - the edges are a property of the finite chain and topological invariant is a property of the bulk/ infinite chain. At the same time, it is really at the crux of topological phases, and this correspondence has been verified over and over experimentally and numerically, even if we don't have a simple answer of why the correspondence exists. I think there is some heavy maths like K-theory that have tried to explain this correspondence but I don't understand it.

One of my supervisors sent me this paper 'Edge states and the bulk-boundary correspondence in Dirac Hamiltonians' https://arxiv.org/abs/1010.2778 which proves why the topological invariants of the bulk lead to edge states but I haven't looked at in depth myself yet.

On a rather random note, there is a really nice set of notes on this website: https://topologicalphases.wordpress.com/lecture-notes/. At the end of Lecture 6 (on the SSH model!), they show why the winding number is equal to the number of edge states for 1D gapped chiral Hamiltonians. It is password protected though. I got access by cold-emailing them and Tomoki Ozawa was nice enough to give me the password (I am a random student and never met them) so maybe you can email them too if you like. They are really nice and clear notes. They consider a special case that has edge states and use the fact that other Hamiltonians with the same winding number can be continuously transformed into the special case (but this also uses homotopy theory which they don't explain in the notes - so the proof of this seemingly simple correspondence is not so simple).

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It is called bulk-boundary correspondence, if the bulk of a system has non-trivial topological invariant, it implies that there will be gapless edge states. This has a rigorous and formal proof with Green's functions. However one can understand this intuitively. The edge of a topological matter means a physical interference of topological matter with trivial insulator we assume that the vacuum is a trivial insulator. Thus you need a gap closing in between that's the reason for the bulk-boundary correspondence.

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