Interplay of topology and interactions in quantum Hall topological insulators: U(1) symmetry, tunable Luttinger liquid, and interaction-induced phase transitions

TL;DR

This paper investigates how electron interactions affect the topological phases and edge states of quantum Hall topological insulators, emphasizing the role of U(1) symmetry in protecting helical Luttinger liquid edges and phase transitions.

Contribution

It establishes the symmetry conditions for topological protection in interacting quantum Hall insulators and provides analytical descriptions of tunable Luttinger liquid parameters.

Findings

U(1) symmetry preservation maintains topological phase and helical Luttinger liquid edges.

Breaking U(1) symmetry leads to interaction-induced phase transitions to trivial gapped phases.

Explicit formulas for Luttinger liquid parameters in the topologically nontrivial phase.

Abstract

We consider a class of {\em quantum Hall topological insulators}: topologically nontrivial states with zero Chern number at finite magnetic field, in which the counter-propagating edge states are protected by a symmetry (spatial or spin) other than time-reversal. HgTe-type heterostructures and graphene are among the relevant systems. We study the effect of electron interactions on the topological properties of the system. We particularly focus on the vicinity of the topological phase transition, marked by the crossing of two Landau levels, where the system is a strongly interacting quantum Hall ferromagnet. We analyse the edge properties using the formalism of the nonlinear $\sigma$-model. We establish the symmetry requirement for the topological protection in this interacting system: effective continuous U(1) symmetry with respect to uniaxial isospin rotations must be preserved. If U(1) symmetry is preserved, the topologically nontrivial phase persists; its edge is a helical Luttinger liquid with highly tunable effective interactions. We obtain explicit analytical expressions for the parameters of the Luttinger liquid. However, U(1) symmetry may be broken, either spontaneously or by U(1)-asymmetric interactions. In either case, interaction-induced transitions occur to the respective topologically trivial phases with gapped edge charge excitations.