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Entanglement Measures for Quasi-Two-Dimensional Fractional Quantum Hall States

We theoretically examine entanglement in fractional quantum hall states, explicitly taking into account and emphasizing the quasi-two-dimensional nature of experimental quantum Hall systems. In particular, we study the entanglement entropy and the entanglement spectrum as a function of the finite layer thickness $d$ of the quasi-two-dimensional system for a number of filling fractions $ν$ in the lowest and the second Landau levels: $ν$ = 1/3, 7/3, 1/2, and 5/2. We observe that the entanglement measures are dependent on which Landau level the electrons fractionally occupy, and find that filling factions 1/3 and 7/3, which are considered to be Laughlin states, weaken with $d$ in the lowest Landau level ($ν$=1/3) and strengthen with $d$ in the second Landau level ($ν$=7/3). For the enigmatic even-denominator $ν=5/2$ state, we find that entanglement in the ground state is consistent with that of the non-Abelian Moore-Read Pfaffian state at an optimal thickness $d$. We also find that the single-layer $ν= 1/2$ system is not a fractional quantum Hall state consistent with the experimental observation. In general, our theoretical findings based on entanglement considerations are completely consistent with the results based on wavefunction overlap calculations.