Identifying the topological order of quantized half-filled Landau levels through their daughter states

TL;DR

This paper links the topological order of half-filled Landau level states to their daughter states, using hierarchy and composite fermion approaches, revealing how the number of chiral Majorana modes influences fractional quantum Hall series.

Contribution

It introduces a method to determine the topological order of half-filled Landau levels by analyzing their daughter states and the role of composite fermions in these quantum Hall states.

Findings

The topological order is uniquely identified by the combination of series above and below half-filling.

A hierarchy and composite fermion approach are used to analyze the states.

A bosonic integer quantum Hall state of composite fermions is crucial for Hall conductivity.

Abstract

Fractional quantum Hall states at a half-filled Landau level are believed to carry an integer number $\mathcal{C}$ of chiral Majorana edge modes, reflected in their thermal Hall conductivity. We show that this number determines the primary series of Abelian fractional quantum Hall states that emerge above and below the half-filling point. On a particular side of half-filling, each series may originate from two consecutive values of $\mathcal{C}$, but the combination of the series above and below half-filling uniquely identifies $\mathcal{C}$. We analyze these states both by a hierarchy approach and by a composite fermion approach. In the latter, we map electrons near a half-filled Landau level to composite fermions at a weak magnetic field and show that a bosonic integer quantum Hall state is formed by pairs of composite fermions and plays a crucial role in the state's Hall conductivity.