# From Quantum Wires to the Chern-Simons Description of the Fractional Quantum Hall Effect

## TL;DR

This paper establishes a direct connection between the microscopic quantum wires approach and the topological Chern-Simons effective theory for the fractional quantum Hall system by bosonizing the former and identifying its fields with the emergent gauge fields.

## Contribution

It provides a novel derivation linking the microscopic fermionic wire model to the topological Chern-Simons description without using bulk-edge correspondence.

## Key findings

- Bosonization of quantum wires system
- Identification of microscopic fields with gauge field components
- Bridge between microscopic and topological descriptions

## Abstract

We show the explicit connection between two distinct and complementary approaches to the fractional quantum Hall system (FQHS): the quantum wires formalism and the topological low-energy effective description given in terms of an Abelian Chern-Simons theory. The quantum wires approach provides a description of the FQHS directly in terms of fermions arranged in an array of one-dimensional coupled wires. In this sense it is usually referred to as a microscopic description. On the other hand, the effective theory has no connection with the microscopic modes, involving only the emergent topological degrees of freedom embodied in an Abelian Chern-Simons gauge field, which somehow encodes the collective motion of the strongly correlated electrons. The basic strategy pursued in this work is to bosonize the quantum wires system and then consider the continuum limit. By examining the algebra of the bosonic operators of the Hamiltonian, we are able to identify the bosonized microscopic fields with the components of the field strength (electric and magnetic fields) of the emergent gauge field. Thus our study provides a bridge between the microscopic physical degrees of freedom and the emergent topological ones, without relying on the bulk-edge correspondence.