Extended Creutz ladder with spin-orbit coupling: a one-dimensional analog of the Kane-Mele model

We construct a topological ladder model, one-dimensional, following the steps which lead to the Kane-Mele model in two dimensions. Starting with a Creutz ladder we modify it so that the gap closure points can occur at either $k = \pi / 2$ or $-\pi/2$. We then couple two such models, one for each spin channel, in such a way that time-reversal invariance is restored. We also add a Rashba spin-orbit coupling term. The model falls in the CII symmetry class. We derive the relevant $2\mathbb{Z}$ topological index, calculate the phase diagram and demonstrate the existence of edge states. We also give the thermodynamic derivation of the quantum spin Hall conductance (St\v{r}eda-Widom). Approximate implementation of this result indicates that this quantity is sensitive to the topological behavior of the model.