Defining the Incompressibility of 2/5 Fractional Quantum Hall State through Gapless Gaffnian State

One of the widely studied fractional quantum Hall states at filling factor 2/5 was portrayed as the so-called Gaffnian state [1], which produces a very high overlap with the exact Coulomb state although it was modeled from non-unitary conformal field theory, thus failing to produce a gaped quantum Hall state. Due to this fact, it was suggested to be a quantum critical point. By recasting the Gaffnian wave function as the inter-flavor pairing of the composite fermions (CFs), we introduce a modified Gaffnian wave function by changing the flux correlations of only two pairs of CFs. A suitable linear combination of these two wave functions has excellent overlap with the exact Coulomb ground state. Agreement of the entanglement spectra corresponding to this mixed wave function and the exact ground-state wave function is found up to much higher energies in comparison to the lone Gaffnian wave function. In contrary to the Gaffnian wave function, this mixed wave function’s overlap with the exact ground state of a hybrid three-body and two-body model Hamiltonian gradually increases with the increase of latter’s weight and eventually becomes almost unity.

[1] S. H. Simon, E. H. Rezayi, N. R. Cooper, and I. Berdnikov, Phys. Rev. B 75, 075317 (2007).