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The experimentally observed fractional quantum Hall states in the lowest Landau level are thought to be described by Laughlin and hierarchy states modeled by Jain's composite fermion wavefunctions. These states have dramatically large overlap with the true Coulomb ground states but the process of flux attachment and projection to the LLL renders them hard to analyze by methods other than Monte-Carlo. Moreover, their large overlap with the true Coulomb ground-state is only empirically understood; this has become most evident recently, when other states at identical filling and shift with the Jain states but exhibiting different topological order, have been found to have competitive overlaps with the true Coulomb ground-states. Although these new states are conjectured to represent critical points, their large overlap with the Jain states (thought to be gapped states) underscores the need to better understand FQH states from a theoretical standpoint.
Identifying the structure of Jain states as "squeezed polynomials", we are able to compute the topological entanglement spectrum of the $\nu= 2/5$ Jain state. We compare it with both the Coulomb ground-state and the non-unitary Gaffnian state. We show that the large overlap of the Jain and Gaffnian states with the Coulomb ground-state is not accidental. Although the Gaffnian state is very close in both overlap and spectral decomposition to Coulomb and Jain, we directly identify the "edge" mode structure of the Coulomb entanglement spectrum and show that it matches the Jain state edge structure as well as that of two $U(1)$ free bosons. |
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