We consider a quaternately generalized Pfaffian QGPf
(J(zi,zj,zk,zl)1)[J(z1,...,zN)]2 in which the square of Vandermonde determinant,
[J(z1,...,zN)]2, implies the upmost Landau level is half filled. This wave function is the unique highest density zero energy state of a special short range interacting Hamiltonian. One can think this quaternate composite fermion liquid as a competing ground state of Moore-Read (MR) Pfaffian state at
ν=5/2. The degeneracy of the quasihole excitations above the QGPf is higher than that of Moore-Read even Read-Rezayi quasiholes. The QGPf is related to a unitary conformal field theory with
Z2×Z2 parafermions in coset space
SU(3)2/U(1)2 . Because of the level-rank duality between
SU(3)2 and
SU(2)3 in conformal field theory, these quasiholes above this QGPf state obeying non-abelian anyonic statistics are expected to support the universal quantum computation at
ν=5/2 as Read-Rezayi quasiholes at
ν=13/5. The edge states of QGPf are very different from those of the Pfaffian's.
The even-denominator fractional quantum Hall (FQH) states, understood as topological superconductors of composite fermions, are predicted to support elementary quasiparticles (QPs) or quasiholes (QHs) obeying non-Abelian braid statistics. Using the microscopic framework of parton theory, we investigate many experimentally observed paired composite-fermion states and find that their elementary QPs and QHs in general bind to form Abelian molecules, thus obscuring non-Abelian braid statistics. An exception is the
ν=1/2 state in wide quantum wells, where our calculations show free QPs or QHs; we thus identify this state as an especially promising candidate for realizing and studying non-Abelian braid statistics. We also indicate how the study of such bound states can help reveal the topological structure of the state.
Non-Abelian excitations are an interesting feature of many fractional quantum Hall phases, including those phases described by the Moore-Read (or Pfaffian) wave function. However, the detection of the non-Abelian quasiparticles is challenging. Here, we consider a system described by the Moore-Read wave function and assume that impurity particles bind to its quasiholes. Then, the angular momentum of the impurities, reflected also by the impurity density, provides a useful witness of the physics of the non-Abelian excitations. By demanding that the impurities are constrained to the lowest Landau level, we are able to write down the corresponding many-body wave function describing both the Moore-Read liquid and the impurities. Through Monte Carlo sampling, we determine the impurity angular momentum, and we show that it suggests a quantum-statistical parameter $\ensuremath{\alpha}=a\ensuremath{\nu}\ensuremath{-}b+P/2$ for the quasiholes, where $\ensuremath{\alpha}$ ranges from 0 for bosons to 1 for fermions. A reasonable agreement with the Monte Carlo results is obtained for
a=1/4,
b=1/8, and
P=0,1 depending on the parity of the particle number in the Moore-Read liquid. This parity dependence of the angular momentum serves as an unambiguous demonstration of the non-Abelian nature of the excitations. In addition to the studies of excitations in the Moore-Read liquid, we also apply our scheme to Laughlin liquids, for which we focus on interacting bosonic impurities. With this, the impurities themselves form Laughlin states, which allows for a study of hierarchical fractional quantum Hall states.