Correlation Lengths and Topological Entanglement Entropies of Unitary and Nonunitary Fractional Quantum Hall Wave Functions

Using the newly developed matrix product state formalism for non-Abelian fractional quantum Hall (FQH) states, we address the question of whether a FQH trial wave function written as a correlation function in a nonunitary conformal field theory (CFT) can describe the bulk of a gapped FQH phase. We show that the nonunitary Gaffnian state exhibits clear signatures of a pathological behavior. As a benchmark we compute the correlation length of a Moore-Read state and find it to be finite in the thermodynamic limit. By contrast, the Gaffnian state has an infinite correlation length in (at least) the non-Abelian sector, and is therefore gapless. We also compute the topological entanglement entropy of several non-Abelian states with and without quasiholes. For the first time in the FQH effect the results are in excellent agreement in all topological sectors with the CFT prediction for unitary states. For the nonunitary Gaffnian state in finite size systems, the topological entanglement entropy seems to behave like that of the composite fermion Jain state at equal filling.

Figure 1

Entanglement entropy $S_A$ for the Moore-Read state in the vacuum and sigma sectors (i.e., with a non-Abelian $\sigma$ quasihole at positions $\pm \infty$) as a function of the cylinder perimeter $L$ and the three largest truncation parameters $P_{\max }$ that can be reached. For large $L$, we observe the saturation due to the finite CFT truncation (the saturation increasing with $P_{\max }.\gamma$ is extracted through a linear fit where we exclude the shaded regions. We consider only perimeters where a convergence of $S_A$ as a function of $P_{\max }$ to be lower than ${10}^{-2}$ has been reached. The extrapolated ${\gamma }_{\text{vac}}$ (${\gamma }_{\text{qh}}$) for the vacuum (sigma) sector is in agreement with the predicted value $\ln \sqrt{8}$ ($\ln 2$). Insets: $S_A$ minus the area law contribution $\alpha L$ for the vacuum (upper left corner) and the sigma (lower right corner) sectors. The value of $\alpha$ which is very close for both topological sectors ($\alpha \simeq 0.22/l_B$), is extracted from the linear fit.

Figure 2

Entanglement entropy $S_A$ for the ${\mathbb{Z}}_3$ Read-Rezayi state in the vacuum sector and with a non-abelian quasihole at each end of the infinite cylinder as a function of the cylinder perimeter $L$ and the three largest truncation parameters $P_{\max }$ that can be reached. We use the same conventions than those of Fig. 1, including for the insets. Note that the lower accuracy on $\gamma$ is due to the data at $P_{\max }=11$. Considering only the two largest truncation parameters would give $\gamma =1.40(3)$. The fitted value of $\alpha \simeq 0.25/l_B$ is also very close for both topological sectors.

Figure 3

Entanglement entropy $S_A$ for the Gaffnian state in the vacuum and sigma sectors as a function of the cylinder perimeter $L$ and the three largest truncation parameters $P_{\max }$ that can be reached. We use the same conventions than those of Fig. 1, including for the insets. The fitted value of $\alpha \simeq 0.20/l_B$ is also very close for both topological sectors. Note that the difference of the entanglement entropies between the two sectors at any perimeter (in the converged region) is always lower than 0.02.

Figure 4

Correlation lengths $\zeta$ of the Laughlin $\nu =1/3$, $\nu =1/5$, and the Moore-Read state for both the vacuum and the quasihole sectors as a function of cylinder perimeter $L$. A linear fit as a function of $1/L$ gives the thermodynamical values ${\zeta }_3/l_B=1.381(1)$ for $\nu =1/3$, ${\zeta }_5/l_B=2.53(7)$ for $\nu =1/5$, ${\zeta }_{\text{vac}}/l_B=2.73(1)$ for the Moore-Read vacuum sector and ${\zeta }_{\text{qh}}/l_B=2.69(1)$ for the Moore-Read quasihole sector.

Figure 5

Correlation lengths $\zeta$ of the Gaffnian state for both the vacuum and the quasihole sectors as a function of cylinder perimeter $L$. A linear fit as a function of $1/L$ for the correlation lengths ${\zeta }_{\text{qh}}$ in the quasihole sector leads to an divergent extrapolated value. In the vacuum sector, due to the level crossing, it is difficult to make any reliable extrapolation.