Particle-Hole Symmetry and the $\nu={5/2}$ Quantum Hall State

TL;DR

This paper explores the implications of particle-hole symmetry in the half-filled Landau level, focusing on the Pfaffian and anti-Pfaffian states, their degeneracy, and experimental relevance in the context of the $ u=5/2$ quantum Hall state.

Contribution

It clarifies the role of particle-hole symmetry in the $ u=5/2$ quantum Hall state and analyzes the competition between Pfaffian and anti-Pfaffian states under Landau level mixing.

Findings

Pfaffian state breaks particle-hole symmetry

Anti-Pfaffian is degenerate with Pfaffian in ideal limit

Landau level mixing influences the favored state

Abstract

We discuss the implications of approximate particle-hole symmetry in a half-filled Landau level in which a paired quantum Hall state forms. We note that the Pfaffian state is not particle-hole symmetric. Therefore, in the limit of vanishing Landau level mixing, in which particle-hole transformation is an exact symmetry, the Pfaffian spontaneously breaks this symmetry. There is a particle-hole conjugate state, which we call the anti-Pfaffian, which is degenerate with the Pfaffian in this limit. We observe that strong Landau level mixing should favor the Pfaffian, but it is an open problem which state is favored for the moderate Landau level mixing which is present in experiments. We discuss the bulk and edge physics of the anti-Pfaffian. We analyze a simplified model in which transitions between analogs of the two states can be studied in detail. Finally, we discuss experimental implications.