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.
Understanding the nature of quasihole excitations, i.e., anyons that have fractional charge and fractional statistics, has been a challenging problem in condensed matter physics. Our theoretical approach to this problem has been to consider a quantum dot containing a few charged particles, Coulomb coupled to the incompressible fluid. The size of the dot and the separation distance of the dot from the incompressible state strongly influences the energetics of the Laughlin quasiholes. Photoluminescence (PL) spectroscopy studies of this system have previously been able to probe these quasiholes that have confirmed our estimates of the Laughlin quasihole energies. Turning to the Pfaffian state, we now observe that such a theoretical approach is also able to provide valuable information about the Pfaffian quasiholes, viz., the energy dispersion, the charge density distribution and the quasihole energy. The energy dispersion of e/4 quasiholes derived here, clearly reflect the interaction between the quantum dot and the incompressible Pfaffian state. PL spectroscopy experiments on the 5/2 Pfaffian-Laughlin state could perhaps shed light on the energetics we found here of these elusive quasiparticles.
The quasihole states of several paired states, the Pfaffian, Haldane-Rezayi, and 331 states, which under certain conditions may describe electrons at filling factor \ensuremath{\nu}=1/2 or 5/2, are studied analytically and numerically in the spherical geometry, for the Hamiltonians for which the ground state are known exactly. We also find all the ground states (without quasiparticles) for these systems in the toroidal geometry. In each case, a complete set of linearly independent functions that are energy eigenstates of zero energy is found explicitly. For fixed positions of the quasiholes, the number of linearly independent states is ${2}^{\mathit{n}\mathrm{\ensuremath{-}}1}$ for the Pfaffian, and ${2}^{2\mathit{n}\mathrm{\ensuremath{-}}3}$ for the Haldane-Rezayi state; these degeneracies are needed if these systems are to possess non-Abelian statistics, and they agree with predictions based on conformal field theory. The dimensions of the spaces of states for each number of quasiholes agree with numerical results for moderate system sizes. The effects of tunneling and of the Zeeman term are discussed for the 331 and Haldane-Rezayi states, as well as the relation to Laughlin states of electron pairs. A model introduced by Ho, which was supposed to connect the 331 and Pfaffian states, is found to have the same degeneracies of zero-energy states as the 331 state, except at its Pfaffian point where it is much more highly degenerate than either the 331 or the Pfaffian. We introduce a modification of the model which has the degeneracies of the 331 state everywhere including the Pfaffian point; at the latter point, tunneling reduces the degeneracies to those of the Pfaffian state. An experimental difference is pointed out between the Laughlin states of electron pairs and the other paired states, in the current-voltage response when electrons tunnel into the edge. An appendix contains results for the permanent state, in which the zero modes can be occupied by composite bosons, rather than by composite fermions as in the other cases; the system is found to have an incipient instability toward a spin-polarized state. \textcopyright{} 1996 The American Physical Society.