Quantum Hall Phase Diagram of Half-filled Bilayers in the Lowest and the Second Orbital Landau Levels: Abelian versus Non-Abelian Incompressible Fractional Quantum Hall States

TL;DR

This study explores the phase diagram of half-filled bilayer quantum Hall systems in the lowest two Landau levels, predicting conditions for Abelian and non-Abelian states and their experimental signatures.

Contribution

It provides new numerical predictions for the occurrence of Abelian and non-Abelian fractional quantum Hall states in bilayer systems at different Landau levels.

Findings

In the lowest Landau level, observed FQHE states are likely Abelian Halperin 331 states.

In the second Landau level, two distinct FQHE branches are predicted, corresponding to Abelian and non-Abelian states.

Possible quantum phase transition between these states could serve as evidence for non-Abelian Moore-Read Pfaffian state.

Abstract

We examine the quantum phase diagram of the fractional quantum Hall effect (FQHE) in the lowest two Landau levels in half-filled bilayer structures as a function of tunneling strength and layer separation, i.e., we revisit the lowest Landau level filling factor 1/2 bilayer problem and make new predictions involving bilayers in the half-filled second Landau level (i.e., filling factor 5/2). Using numerical exact diagonalization we investigate the important question of whether this system supports a FQHE described by the non-Abelian Moore-Read Pfaffian state in the strong tunneling regime. In the lowest Landau level, we find that although in principle, increasing (decreasing) tunneling strength (layer separation) could lead to a transition from the Abelian two-component Halperin 331 to non-Abelian one-component Moore-Read Pfaffian state, the FQHE excitation gap is relatively small in the lowest Landau level Pfaffian regime--we establish that all so far observed FQHE states in half-filled lowest Landau level bilayers are most likely described by the Abelian Halperin 331 state. In the second Landau level we make the prediction that bilayer structures would manifest two distinct branches of incompressible FQHE corresponding to the Abelian 331 state (at moderate to low tunneling and large layer separation) and the non-Abelian Moore-Read Pfaffian state (at large tunneling and small layer separation). The observation of these two FQHE branches and the possible quantum phase transition between them will be compelling evidence supporting the existence of the non-Abelian Moore-Read Pfaffian state in the second Landau level. We discuss our results in the context of existing experiments and theoretical works.