Composite helical edges from Abelian fractional topological insulators

TL;DR

This paper develops a systematic framework for composite Abelian helical edge states with fractionalization, analyzing their phases, conductance properties, and responses to magnetic fields, with implications for recent experiments on fractional topological insulators.

Contribution

It introduces a comprehensive theoretical model for composite $(1+1/n)$ Abelian helical edges, including phase diagrams, conductance calculations, and experimental signatures, extending understanding of fractional topological insulators.

Findings

Identification of various phases including localized insulator and drag phases.

Calculation of edge conductance showing unconventional values like $(1-1/n) e^2/h$.

Prediction of magnetic field effects on edge conductance for experimental tests.

Abstract

We study an interacting composite $(1+1/n)$ Abelian helical edge state made of a regular helical liquid carrying charge $e$ and a (fractionalized) helical liquid carrying charge $e/n$. A systematic framework is developed for these composite $(1+1/n)$ Abelian helical edge states with $n=1,2,3$. For $n=2$, the composite edge state consists of a regular helical Luttinger liquid and a fractional topological insulator (the Abelian $Z_4$ topological order) edge state arising from half-filled conjugated Chern bands. The composite edge state with $n=2$ is pertinent to the recent twisted MoTe$_2$ experiment, suggesting a possible fractional topological insulator with conductance $\frac{3}{2}\frac{e^2}{h}$ per edge. Using bosonization, we construct generic phase diagrams in the presence of $weak$ Rashba spin-orbit coupling. In addition to a phase of free bosons, we find a time-reversal symmetry-breaking localized insulator, two perfect positive drag phases, a perfect negative drag phase (for $n=2,3$), a time-reversal symmetric Anderson localization (only for $n=1$), and a disorder-dominated metallic phase analogous to the $\nu=2/3$ disordered fractional quantum Hall edges (only for $n=3$). We further compute the two-terminal edge-state conductance, the primary experimental characterization for the (fractional) topological insulator. Remarkably, the negative drag phase gives rise to an unusual edge-state conductance, $(1-1/n)\frac{e^2}{h}$, not directly associated with the filling factor. We further investigate the effect of an applied in-plane magnetic field. For $n>1$, the applied magnetic field can result in a phase with edge-state conductance $\frac{1}{n}\frac{e^2}{h}$, providing another testable signature. Our work establishes a systematic understanding of the composite $(1+1/n)$ Abelian helical edge, paving the way for future experimental and theoretical studies.