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

TL;DR

This work addresses the existence and nature of non-Abelian fractional quantum Hall states at filling $\nu=\frac{3}{4}$, analyzing two complementary theoretical constructions—particle-hole conjugation of $\nu=\frac{1}{4}$ Moore-Read-type states and a composite-fermion picture with effective $\nu^{*}= -\frac{3}{2}$—and validating them against bilayer graphene simulations. It identifies three candidate non-Abelian orders ($\frac{3}{4}$-Pf, $\frac{3}{4}$-aPf, and $\frac{3}{4}$-PHS-Pf) and argues their topological equivalence, with graviton spectral functions providing diagnostic signatures of the underlying order. Through exact diagonalization in a BLG model including LL mixing, the authors observe a fragile but robust 12-fold quasi-degenerate ground-state manifold on the torus, whose prominence grows with LL mixing and is accompanied by a pair of chiral graviton modes (one negative, one positive chirality) consistent with the $\frac{3}{4}$-aPf universality class. The findings support a Moore-Read-type non-Abelian order at $\nu=\frac{3}{4}$ in BLG, with implications for GaAs hole systems and other van der Waals platforms, and underscore LL mixing as a crucial ingredient for stabilizing such states.

Abstract

Fractional quantum Hall states have been observed at filling factor $ν=3/4$ 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 $ν=3/4$, which exhibit $12$ fold ground state degeneracy on the torus. The properties of $ν=3/4$ states can be analyzed using two complementary approaches. In the first one, they are treated as particle-hole conjugate of $ν=1/4$ 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 $ν=1/2$ Moore-Read type states. For bilayer graphene with appropriate Landau level mixing, numerical calculations found $12$ quasi-degenerate ground states on the torus at $ν=3/4$. 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.

Figures (3)

• Figure 1: (a) Schematics of the $\nu=3/4$ states constructed by the first approach. (b) Parton interpretation of the $\nu=1/4$ states. (c) Schematics of the $\nu=3/4$ states constructed by the second approach. (d) Schematics of the microscopic model used in numerical calculations.
• Figure 2: (a-c) Energy spectra of the system with $18$ electrons and $N_{\phi}=24$. (d-e) Energy spectra of the system with $24$ electrons and $N_{\phi}=32$. The parameter $M_{c}$ is given in each panel and other parameters are given in the main text.
• Figure 3: (a-b) Chiral graviton spectral functions of the three quasi-degenerate ground states in Fig. \ref{['Figure1']} (c). (c-d) Chiral graviton spectral functions of the three quasi-degenerate ground states in Fig. \ref{['Figure1']} (e). When there are two states in the same momentum sector, they are distinguished by the label $a$ and $b$. The chirality is negative in panels (a,c) and positive in panels (b,d).