Argument for number of edge states as topological invariant for SSH model

Question

I am currently reading the book "A short introduction to Topological insulators" by Asboth etal.
In the first chapter on SSH model, they argue (see sec 1.5.3) that number of edge states is a topological invariant for SSH. I have some specific issues in the argument that follows:-

1. The first line reads: "Consider energy eigenstates at left end of a ..." - What exactly is meant by left end here? Do they mean states localised on the left end?
   This seems a bit ambiguous since their plot in fig.1.4 highlights two edge states, which are hyrbidization of left and right localised states. So in that case, what would "energy eigenstates at left end" correspond to?

2. "Number of 0 energy states is finite because of gap in the bulk" - Why? (this is to be true even when $N \rightarrow \infty$ as stated in beginning of their argument)

3. At the end, why is $N_A-N_B$ called the "net" number of edge states on sublattice A at left end? From what I understand till now, $N_A$ states have support on sublattice A while $N_B$ states have it on sublattice B. So why call their difference as "net" states on sublattice A?

Thanks in advance for the help ;)

Answer 1

1. Yes, we meant states that are localized at the left end. For any finite chain, edge states at the two ends would be at the same energy (0), and therefore always hybridize, and true eigenstates are there even and odd superpositions. However, increasing the length, the hybridization becomes exponentially small. That is why we say later in that sentence, that you should take the limit of infinite length.

2. Because of the bulk gap, any eigenstate at 0 energy has to have a wavefunction that decays exponentially towards the bulk. Thus any 0 energy states have to be confined to the vicinity of the edge, and the size of "edge region" that can support such eigenstates is in a sense finite, and does not increase as you increase the system size.

3. This can be called a "net" number, because increasing N_B is like decreasing N_A: if you introduce a new 0 energy state on sublattice B, it can hybridize with a state on sublattice A, and together they can leave the 0 energy subspace.

Hope this helps.