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The PH-Pfaffian state having 1/2 central charge is consistent with thethermal Hall conductance measurement of 5/2 fractional quantum Hall system, butlacks support from the existing numerical results. In this paper we propose anew state described by a wavefunction obtained by multiplying a Hafnian factorto a PH-Pfaffian wavefunction. We call this new state the Hafnian PH-Pfaffianstate. In spherical geometry, the Hafnian PH-Pfaffian state has the samemagnetic flux number as the Pfaffian state, allowing a direct numericalcomparison between the two states. Results of exact diagonalization of finitesystems in the second Landau level show that the overlap of the exact groundstate with the Hafnian PH-Pfaffian state exceeds that with the Pfaffian statewhen the short range component of the Coulomb interaction increases to acertain level, lending a numerical support to the Hafnian PH-Pfaffian state. Wefurther show the Hafnian PH-Pfaffian state is mathematically identical to thenewly proposed compressed PH-Pfaffian state [arxiv: 2001.01915 (2020)] formedby "compressing" the PH-Pfaffian state with two flux quanta removed to createtwo abelian Laughlin type quasiparticles of the maximum avoidance from oneanother. As the result we argue that the Hafnian PH-Pfaffian state has the samecentral charge as the PH-Pfaffian state, and is therefore consistent with thethermal Hall conductance measurement. Finally we present numerical results ontwo new wavefunctions formed by increasing the relative angular momentum by twofor each of the paired composite fermions of the PH-Pfaffian state.

A Hafnian PH-Pfaffian State for 5/2 Quantum Hall Effect