Phase diagram of two interacting helical states

TL;DR

This paper analyzes the phase diagram of two coupled helical edge states in quantum spin Hall insulators, revealing conditions for gapped phases, topological properties, and impurity protection.

Contribution

It introduces a detailed theoretical framework for understanding the interaction-driven phases and topological features of coupled helical edge modes.

Findings

Relative mode becomes gapped with tunneling.

Spin-nematic phase arises when intra-edge interactions dominate.

Topological spin-density wave phase hosts zero modes.

Abstract

We consider two coupled time reversal invariant helical edge modes of the same helicity, such as would occur on two stacked quantum spin Hall insulators. In the presence of interaction, the low energy physics is described by two collective modes, one corresponding to the total current flowing around the edge and the other one describing relative fluctuations between the two edges. We find that quite generically, the relative mode becomes gapped at low temperatures, but only when tunneling between the two helical modes is non-zero. There are two distinct possibilities for the gapped state depending on the relative size of different interactions. If the intra-edge interaction is stronger than the inter-edge interaction, the state is characterised as a spin-nematic phase. However in the opposite limit, when the interaction between the helical edge modes is strong compared to the interaction within each mode, a spin-density wave forms, with emergent topological properties. Firstly, the gap protects the conducting phase against localization by weak nonmagnetic impurities; and secondly the protected phase hosts localized zero modes on ends of the edge that may be created by sufficiently strong non-magnetic impurities.