Non-Abelian fractional quantum Hall states at filling factor 3/4

TL;DR

This paper explores non-Abelian fractional quantum Hall states at filling factor 3/4, revealing their theoretical properties, possible realizations, and numerical evidence for their existence in bilayer graphene, highlighting their topological order and ground state degeneracy.

Contribution

It introduces a comprehensive analysis of non-Abelian states at 3/4 filling, combining bootstrap theory, particle-hole conjugation, and composite fermion mapping, with numerical validation.

Findings

Non-Abelian states at 3/4 filling exhibit 12-fold ground state degeneracy.

Numerical calculations in bilayer graphene support the existence of these states.

Chiral graviton spectral functions show characteristic low and high energy peaks.

Abstract

Fractional quantum Hall states have been observed at filling factor $\nu=3/4$ in GaAs hole system and bilayer graphene. In theoretical bootstrap analysis, it was revealed that non-Abelian topological orders with Ising anyons can be realized at $\nu=3/4$, which exhibit $12$ fold ground state degeneracy on the torus. The properties of $\nu=3/4$ states can be analyzed using two complementary approaches. In the first one, they are treated as particle-hole conjugate of $\nu=1/4$ Moore-Read types states. In the second one, they are mapped to composite fermions with reverse flux attachment at effective filling factor $3/2$, whose integral part realizes an integer quantum Hall state and the fractional part realizes $\nu=1/2$ Moore-Read type states. For bilayer graphene with appropriate Landau level mixing, numerical calculations found $12$ quasi-degenerate ground states on the torus at $\nu=3/4$. Chiral graviton spectral functions of these states have one low energy peak with negative chirality and one high energy peak with positive chirality. This points to a specific member of the Moore-Read type states and agrees with the deduction based on daughter states.