# How to describe SSH chain with odd number of sites?

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?

• Two minor suggestions for an edit. (1) Define “SSH” for folks who may not know the acronym. (2) Use \exp (with the backslash) in your typeset expressions.
– rob
Commented Oct 24, 2021 at 16:02