Entanglement Entropy in Fermionic Laughlin States

We present analytic and numerical calculations on the bipartite entanglement entropy in fractional quantum Hall states of the fermionic Laughlin sequence. The partitioning of the system is done both by dividing Landau-level orbitals and by grouping the fermions themselves. For the case of orbital partitioning, our results can be related to spatial partitioning, enabling us to extract a topological quantity (the “total quantum dimension”) characterizing the Laughlin states. For particle partitioning we prove a very close upper bound for the entanglement entropy of a subset of the particles with the rest, and provide an interpretation in terms of exclusion statistics.

Figure 1

Entanglement entropies with orbital partitioning ($S_{l_A}$), for various $N$ and $m$. Inset: maximum $S_{l_A}$ values plotted against square root of positions $l_A^{*}$ of the maximum, for $m=3$ wave functions of various sizes.

Figure 2

Entanglement entropies in $m=3$ states extrapolated to the thermodynamic limit. Dashed line is a fit to $-\gamma +c_1\sqrt{l_A}$ giving more weight to the higher points. Inset on right plots $S_{l_A}$ against $N$, for $l_A=1$, 3, 6, 10. With wave functions available up to $N=10$, the $N\to \infty$ extrapolation is reliable for small $l_A$ but is already difficult for $l_A=10$ because the $S_{l_A}(N)$ curve is not nearly flat yet.

Figure 3

Entanglement entropy of two particles ($A$) with the rest ($B$). Dots are numerical exact values; dashed curves are naïve estimates [Eq. (2)]; full curves are improved bounds taking multiplet structure into account.