How to describe SSH chain with odd number of sites?

Question

Usually when we discuss SSH(Su-Schrieffer–Heeger) chain, we discuss a chain with 2N atoms, with v the intra-cell coupling and w the inter-cell coupling. When N is infinite, the system becomes bulk, with Hamiltonian $$H(k)=\begin{pmatrix} 0&v+w\,\exp(-ik)\\ v+w\,\exp(ik)&0 \end{pmatrix}$$ We can calculate the winding number of H(k) for different parameter v and w, which would be either 1 (v<w) or 0 (v>w). This winding number predicts the pair of edge states at the Fermi surface when the chain is cut finite, i.e. when N is finite.

What happens when we consider a chain with 2N+1 atoms, with N finite? This chain does not have a bulk. Straight forward calculation shows that arbitrary value of v and w, this odd-site SSH chain always have one zero energy edge state, but not a pair? Is there a topological explanation for this, i.e. is there anything resembling the winding number for this chain?

Answer 1

There a discussion of the odd site problem in my ancient Zero modes, boundary conditions and anomalies on the lattice and in the continuum, Annals of Physics 155 (1), 56-84. It does not mention winding numbers and it rather simple minded by today's standards though.