$\mathbb Z_2$ or $\mathbb Z$ invariant for the SSH model

Question

I am trying to understand topological insulators and topological invariant. The SSH (Su-Schrieffer-Heeger) model is often invoked as a protoypical topological insulator in 1D that carries localized zero modes at the edge. In every single treatment I could find, people compute winding numbers or Zak phases that can have one of two possible values. Thus, they are $\mathbb Z_2$ invariants, right?

Then, often the classification of topological insulators from symmetries is often discussed, and a "periodic table" is presented. (For instance: https://topocondmat.org/w8_general/classification.html). The SSH model falls into class AIII or BDI, depending on whether one considers the electronic or the mechanical case (as in Kane & Lubensky 2013, Topological Boundary Modes in Isostatic Lattices). However, in $d=1$, these periodic tables predict a $\mathbb Z$ invariant, not a $\mathbb Z_2$ one!

So what is it that I am not understanding here? Is the invariant from the periodic table a different one? What is the $\mathbb Z$ invariant for the SSH model then? Or am I reading the table wrong?

This post imported from StackExchange Physics at 2018-06-19 08:54 (UTC), posted by SE-user henrikr