Quasi 1D insulators with strong spin-orbital interaction

Question

We know that the spin-1 chain realizes the Haldane phase which is an example of symmetry protected topological (SPT) phases (ie short-range entangled phases with symmetry). The Haldane phase is protected by the $SO(3)$ spin rotation symmetry. If we change the symmetry, we may obtain other possible SPT phases. This motivates us to ask the following question:

What are good material examples of quasi 1D insulators with strong spin-orbital interaction?

There are large $U$ Mott insulators and there are small $U$ band insulators. Here, we are interested in both, and like to see examples for both cases.

Answer 1

I can provide an example for bosonic models.

\begin{eqnarray} \mathcal{H} & = & \mathcal{K} + \mathcal{T}_\text{soc} +\frac{U}{2}\sum_{i\tau} \hat n_{i\tau}( \hat n_{i\tau}-1) \nonumber \\ & & + U^{\prime} \sum_i \hat n_{i\uparrow} \hat n_{i\downarrow} + V\sum_{i\tau} \hat{n}_{i\tau}\hat{n}_{i+1\tau} \nonumber \\ & & +V^\prime\sum_{i\tau} \hat{n}_{i\tau} \hat{n}_{i+1\bar{\tau}} -\mu\sum_{i}\hat n_{i}, \label{HSOC} \end{eqnarray} where \begin{equation} \mathcal{T}_\text{soc} = -\lambda\sum_i(\hat{c}^{\dagger}_{i\uparrow}\hat{c}_{i+1\downarrow} -\hat{c}^{\dagger}_{i\downarrow}\hat{c}_{i+1\uparrow})+h.c., \end{equation} is the spin-orbit coupling, $\mathcal{K}$ is the hopping term.