Quantum Berezinskii-Kosterltz-Thouless Transition for Topological Insulator

TL;DR

This paper studies a topological insulator edge system under magnetic and superconducting influences, revealing a quantum BKT transition, the behavior of Majorana modes, and the absence of a Majorana-Ising transition.

Contribution

It maps the interacting helical liquid to an XYZ spin chain, derives RG equations, and analyzes the quantum BKT transition and phase behavior in detail.

Findings

Identification of quantum BKT transition in the model

Evidence of gapless helical Luttinger liquid phase

No Majorana-Ising transition occurs in the system

Abstract

We consider the interacting helical liquid system at the one-dimensional edge of a two-dimensional topological insulator, coupled to an external magnetic field and s-wave superconductor and map it to an XYZ spin chain system. This model undergoes quantum Berezinskii-Kosterlitz-Thouless (BKT) transition with two limiting conditions. We derive the renormalization group (RG) equations explicitly and also present the flow lines behavior. We also present the behavior of RG flow lines based on the exact solution. We observe that the physics of Majorana fermion zero modes and the gaped Ising-ferromagnetic phase, which appears in a different context. We observe that the evidence of gapless helical Luttinger liquid phase as a common non-topological quantum phase for both quantum BKT transitions. We explain analytically and physically that there is no Majorana-Ising transition. In the presence of chemical potential, the system shows the commensurate to incommensurate transition.