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

TL;DR

This paper introduces a one-dimensional topological ladder model with spin-orbit coupling, mimicking the Kane-Mele model, and demonstrates its topological phases, edge states, and spin Hall conductance, providing insights into quantum spin Hall effects.

Contribution

It constructs a 1D analog of the Kane-Mele model with spin-orbit coupling and analyzes its topological properties and edge states, extending topological insulator concepts to one dimension.

Findings

The model exhibits topological phases with edge states.

The topological index is derived and calculated.

The quantum spin Hall conductance is shown to be sensitive to topology.

Abstract

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.