Pfaffian paired states for half-integer fractional quantum Hall effect

TL;DR

This paper reviews Pfaffian paired states in the fractional quantum Hall effect, focusing on their realization, competition, and the influence of Landau level mixing using field-theoretical methods.

Contribution

It introduces a field-theoretical approach to analyze Pfaffian states, including the influence of Landau level mixing and the generation of interaction pseudo-potentials.

Findings

Pfaffian and anti-Pfaffian states are favored with weak Landau level mixing.

PH Pfaffian state requires non-perturbative treatment of Landau levels.

Field-theoretical methods can generate characteristic interaction pseudo-potentials.

Abstract

In this review the physics of Pfaffian paired states, in the context of fractional quantum Hall effect, is discussed using field-theoretical approaches. The Pfaffian states are prime examples of topological ($p$-wave) Cooper pairing and are characterized by non-Abelian statistics of their quasiparticles. Here we focus on conditions for their realization and competition among them at half-integer filling factors. Using the Dirac composite fermion description, in the presence of a mass term, we study the influence of Landau level mixing in selecting a particular Pfaffian state. While Pfaffian and anti-Pfaffian are selected when Landau level mixing is not strong, and can be taken into account perturbatively, the PH Pfaffian state requires non-perturbative inclusion of at least two Landau levels. Our findings, for small Landau level mixing, are in accordance with numerical investigations in the literature, and call for a non-perturbative approach in the search for PH Pfaffian correlations. We demonstrated that a method based on the Chern-Simons field-theoretical approach can be used to generate characteristic interaction pseudo-potentials for Pfaffian paired states.