Explicit mapping between a 2D quantum Hall system and a 1D Luttinger liquid, I. Luttinger parameters

TL;DR

This paper establishes a detailed microscopic mapping between a 2D quantum Hall system confined to a narrow channel and a 1D Luttinger liquid, analyzing how interactions influence system parameters.

Contribution

It provides a microscopic derivation of the mapping and clarifies the relationship between 2D quantum Hall parameters and 1D Luttinger liquid parameters, including interaction effects.

Findings

The velocity of the current remains unchanged by interactions.

Interaction parameters are renormalized in the mapping.

The model offers insights into low-energy behavior of quantum Hall edges.

Abstract

We study a simple model of a quantum Hall system with the electrons confined to a linear, narrow channel. The system is mapped to a 1D system which in the low-energy approximation has the form of a Luttinger liquid with different interactions between particles of equal and of opposite chiralities. We study this mapping at the microscopic level, and discuss the relation between the parameters of the 2D system and the corresponding 1D Luttinger liquid parameters. We focus in particular on how the parameters are renormalized by the electron interactions and show that the velocity parameter of the current is not modified by the interaction.