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

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.

💡 Research Summary

The paper investigates the fractional quantum Hall (FQH) state observed at filling factor ν = 3/4 in GaAs hole systems and bilayer graphene (BLG). Using a bootstrap analysis, the authors argue that the ν = 3/4 state must realize a non‑Abelian topological order (TO) with Ising anyons, which on a torus manifests as a twelve‑fold ground‑state degeneracy. They develop two complementary theoretical constructions that lead to the same TO.

The first construction starts from the well‑studied ν = 1/2 Moore‑Read Pfaffian family. By multiplying a Moore‑Read wave function (or its anti‑Pfaffian or particle‑hole symmetric variant) with a Jastrow factor ∏(z_i − z_j)^2 they obtain a ν = 1/4 state. Particle‑hole conjugation then yields a ν = 3/4 state, denoted Ψ_I^{3/4}=PH Ψ_I^{1/4}. This approach naturally produces three candidate states—3/4‑Pfaffian, 3/4‑anti‑Pfaffian, and 3/4‑PH‑symmetric Pfaffian—each characterized by a distinct chiral central charge (c = 3/2, −1/2, 1/2 respectively) and a specific shift on the sphere.

The second construction employs composite‑fermion (CF) theory with reverse flux attachment. Electrons bind two flux quanta to become CFs; the effective filling factor for CFs is ν* = −3/2, meaning they experience a negative effective magnetic field. The lowest CF Landau level is completely filled (an integer quantum Hall state), while the second CF Landau level is half‑filled and forms a Moore‑Read‑type paired state. The resulting wave function, Ψ_II^{3/4}=h Ψ_MR^{1/2}⊗Ψ_IQH^{1}·∏(z_i − z_j)^2, includes a complex conjugation h reflecting the reversed field. This picture reproduces the same electron‑number, flux‑number, shift, and chiral central charge as the particle‑hole construction, establishing a one‑to‑one correspondence between the two approaches.

To distinguish among the three candidates, the authors turn to the spectrum of chiral gravitons—spin‑2 neutral excitations associated with fluctuations of the intrinsic metric of the quantum Hall fluid. In the Moore‑Read family, the Pfaffian exhibits a single negative‑chirality graviton, the anti‑Pfaffian a single positive‑chirality graviton, while the PH‑symmetric Pfaffian shows both chiralities. For ν > 1/2 states, theory predicts a low‑energy graviton of one chirality and a higher‑energy graviton of the opposite chirality. Thus, observing both peaks would point to the anti‑Pfaffian (or a related member) for ν = 3/4.

The numerical part focuses on BLG, where the authors construct a simplified four‑band single‑particle Hamiltonian based on the Slonczewski‑Weiss‑McClure parameters (γ0, γ1, γ3, γ4, δ) and include a displacement field that polarizes spin and valley. They truncate the Hilbert space to two effective Landau levels, Φ0 and Φ1, with an adjustable inter‑level spacing ℏΩ. Electrons interact via a screened Coulomb potential V_sc(q)= (e^2/4πϵℓ_B)·(2πℓ_B/|q|)·tanh(|q|d), where d is the screening length. Exact diagonalization is performed on a rectangular torus with N_ϕ magnetic flux quanta.

Scanning over interaction parameters, the authors find a regime where twelve quasi‑degenerate ground states appear, confirming the bootstrap prediction of a 12‑fold degeneracy. They compute the graviton spectral function and observe two distinct peaks: a low‑energy peak with negative chirality and a higher‑energy peak with positive chirality. This pattern matches the expected signature of the anti‑Pfaffian (or its anti‑Pfaffian‑like daughter) state.

In summary, the paper provides (i) a bootstrap‑derived constraint that any ν = 3/4 state with 12‑fold torus degeneracy must be an Ising‑anyonic non‑Abelian TO; (ii) two theoretically equivalent constructions (particle‑hole conjugation of ν = 1/4 Moore‑Read states and reverse‑flux‑attachment CF theory) that generate the same set of candidate wave functions; (iii) a detailed analysis of graviton excitations as a bulk probe to differentiate among the candidates; and (iv) concrete numerical evidence from a BLG model that the anti‑Pfaffian‑type TO is realized in realistic experimental conditions. The work thus strengthens the case for non‑Abelian anyons at ν = 3/4 and highlights the potential of such states for topological quantum computation.