There are several possible theoretically allowed non-Abelian fractional quantum Hall (FQH) states that could potentially be realized in one- and two-component FQH systems at total filling fraction $\ensuremath{\nu}=n+2/3$, for integer
n. Some of these states even possess quasiparticles with non-Abelian statistics that are powerful enough for universal topological quantum computation, and are thus of particular interest. Here we initiate a systematic numerical study, using both exact diagonalization and variational Monte Carlo, to investigate the phase diagram of FQH systems at total filling fraction $\ensuremath{\nu}=n+2/3$, including in particular the possibility of the non-Abelian
Z4 parafermion state. In $\ensuremath{\nu}=2/3$ bilayers we determine the phase diagram as a function of interlayer tunneling and repulsion, finding only three competing Abelian states, without the
Z4 state. On the other hand, in single-component systems at $\ensuremath{\nu}=8/3$, we find that the
Z4 parafermion state has significantly higher overlap with the exact ground state than the Laughlin state, together with a larger gap, suggesting that the experimentally observed $\ensuremath{\nu}=8/3$ state may be non-Abelian. Our results from the two complementary numerical techniques agree well with each other qualitatively.
Topological quantum states with non-Abelian Fibonacci anyonic excitations are widely sought after for the exotic fundamental physics they would exhibit, and for universal quantum computing applications. The fractional quantum Hall (FQH) state at a filling factor of ν=12/5 is a promising candidate; however, its precise nature is still under debate and no consensus has been achieved so far. Here, we investigate the nature of the FQH ν=13/5 state and its particle-hole conjugate state at 12/5 with the Coulomb interaction, and we address the issue of possible competing states. Based on a large-scale density-matrix renormalization group calculation in spherical geometry, we present evidence that the essential physics of the Coulomb ground state (GS) at ν=13/5 and 12/5 is captured by the k=3 parafermion Read-Rezayi state (RR_{3}), including a robust excitation gap and the topological fingerprint from the entanglement spectrum and topological entanglement entropy. Furthermore, by considering the infinite-cylinder geometry (topologically equivalent to torus geometry), we expose the non-Abelian GS sector corresponding to a Fibonacci anyonic quasiparticle, which serves as a signature of the RR_{3} state at 13/5 and 12/5 filling numbers.
The generalized exclusion statistics (GES) parameter
g characterizes the fractionalization of anyonic excitations in fractional quantum Hall (FQH) states. While
g is phenomenologically linked to the filling factor
ν in Laughlin states, a universal derivation from the underlying topological field theory for arbitrary Abelian and non-Abelian states remains an open theoretical target. Furthermore, whether this topologically derived
g can quantitatively predict the shot-noise Fano factor
F measured in quantum point contact (QPC) tunneling experiments has not been rigorously established across general FQH phases. Here, we derive
g directly from the topological spin
θ=e2πih, where
h is the conformal dimension of the corresponding conformal field theory (CFT) primary field. For Abelian anyons,
g=2h, while non-Abelian anyons yield a matrix form. Using the Keldysh formalism for the QPC tunneling Hamiltonian, we show this topologically derived
g dictates the tunneling density of states exponent and predicts the Fano factor
F=gν in the weak backscattering regime. For Laughlin states at
ν=1/m, this yields
F=ν, matching fractional charge
e∗=νe. For non-Abelian states, the derived
g matrix predicts a modified Fano factor revealing topological spin contributions to nonequilibrium noise. ## 1. Introduction The fractional quantum Hall (FQH) effect provides a physical realization of topological phases of matter hosting anyonic excitations. These excitations obey generalized exclusion statistics (GES), a framework introduced to generalize the Pauli exclusion principle to particles with fractional braiding statistics. In a quantum point contact (QPC) geometry, the tunneling of these anyons generates shot noise, characterized by the Fano factor
F (the ratio of noise power to twice the Poissonian current). While the connection between GES and the filling factor
ν is well-established for Laughlin states, a universal deriv
We compute the physical properties of non-Abelian fractional quantum Hall (FQH) states described by Jack polynomials at general filling nu = k/r. For r = 2, these states are the Zk Read-Rezayi parafermions, whereas for r > 2 they represent new FQH states. The r = k+1 states, multiplied by a Vandermonde determinant, are a non-Abelian alternative construction of states at fermionic filling 2/5,3/7,4/9,.... We obtain the thermal Hall coefficient, the quantum dimensions, the electron scaling exponent, and the non-Abelian quasihole propagator. The properties of the r > 2 Jack polynomials indicate they are correlators of fields of nonunitary conformal field theories (CFT), but the CFT-FQH connection fails when invoked to compute physical properties such as the quasihole propagator. The quasihole wave function, written as a coherent state representation of Jack polynomials, has an identical structure for all non-Abelian states.
We use exact numerical diagonalization in the search of fractional quantum Hall states with non-Abelian quasiparticle statistics. For the (most promising) states in a partially filled second Landau level, the search is narrowed to the range of filling factors 7/3 < v e < 8/3. In this range, the analysis of energy spectra and correlation functions, calculated including finite width and Landau level mixing, supports the prominent non-Abelian candidates at v e = 5/2 (paired Moore–Read "pfafian" state) and 12/5 (clustered Read–Rezayi "parafermion" state). Outside of this range, the noninteracting composite fermion model with four attached flux quanta is validated, yielding the family of quantum liquids with fractional, but Abelian statistics. The borderline v e = 7/3 state is shown to be adiabatically connected to the Laughlin liquid, but its short-range correlations are significantly different.
Quasiparticle tunneling between two counter-propagating edges through point contacts could provide information on the statistics of the quasiparticles. Previous study on a disk found a scaling behavior by varying the tunneling distance. It was found that in the limit with zero tunneling distance, the Abelian quasiparticles tunneling obey the scaling analysis while the non-Abelian quasiparticles exhibit some non-trivial behaviors on the scaling exponents. Because of the limitation of disk geometry, we put the fractional quantum Hall (FQH) state on the surface of a cylinder which has a larger tunable tunneling distance than that on disk by varying the aspect ratio
γ. We analyze the scaling behavior of the quasiholes, especially the non-Abelian quasiholes in the Read-Rezayi
Zk parafermion states. We aim to address the existance of the anomalous correction of the scaling parameter in the long tunneling distance.