Quaternate generalization of Pfaffian state at $\nu =5/2$

Abstract

We consider a quaternately generalized Pfaffian QGPf$(\frac{1}{J(z_i,z_j,z_k,z_l)})[J(z_1,...,z_N){]}^2$ in which the square of Vandermonde determinant, $[J(z_1,...,z_N){]}^2$, 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 $\nu =5/2$. 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 $Z_2\times Z_2$ 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 $\nu =5/2$ as Read-Rezayi quasiholes at $\nu =13/5$. The edge states of QGPf are very different from those of the Pfaffian's.

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