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

TL;DR

This paper introduces a quaternate generalization of the Pfaffian state at filling factor 5/2, proposing a new wave function with unique topological properties and potential for universal quantum computation.

Contribution

It presents a novel quaternate Pfaffian wave function as a competing ground state at 5/2, with distinct quasihole degeneracies and non-abelian statistics linked to a specific conformal field theory.

Findings

The QGPf state is the highest density zero energy state of a particular Hamiltonian.

Quasiholes in the QGPf exhibit higher degeneracy than Moore-Read quasiholes.

QGPf quasiholes support non-abelian anyonic statistics suitable for universal quantum computation.

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.