PFfaffian, Gaffnian and Haffnian states : Quantum Computations

Abstract

The number of particles in a cluster of electrons determines the angular momentum which is used to define the filling factor of the lowest Landau level. A fuzz factor is added to it so that it gives the desired denominators. Laughlin state is obtained for any value of the fuzz factor, p. For g = 2, p = 4 we get the half filled Landau level, the state which is called Haffnian. For p = 2 there are always two particles which come from non Abelian determinant called pFaffian. For g = 2, p = 3 the state is called Gaffnian. In all cases, the ground state belongs to non physical many body Hamiltonians. The composite fermion wave function is not the ground state of any known single Hamiltonian. From the angular momentum, we construct the filling factor and then look for the wave function and then for the Hamiltonian. In this approach the Hamiltonians found are unrealistic. It is possible to make the filling factors agree with the experimental value but then Hamiltonians are not the usual type.