Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states

We consider the trial wave functions for the fractional quantum Hall effect that are given by conformal blocks, and construct their associated edge excited states in full generality. The inner products between these edge states are computed in the thermodynamic limit, assuming generalized screening (i.e., short-range correlations only) inside the quantum Hall droplet and using the language of boundary conformal field theory (boundary CFT). These inner products take universal values in this limit: they are equal to the corresponding inner products in the bulk two-dimensional chiral CFT which underlies the trial wave function. This is a bulk/edge correspondence; it shows the equality between equal-time correlators along the edge and the correlators of the bulk CFT up to a Wick rotation. This approach is then used to analyze the entanglement spectrum of the ground state obtained with a bipartition $A\cup B$ in real space. Starting from our universal result for inner products in the thermodynamic limit, we tackle corrections to scaling using standard field-theoretic and renormalization-group arguments. We prove that generalized screening implies that the entanglement Hamiltonian $H_E=-\ln {\rho }_A$ is isospectral to an operator that is local along the cut between $A$ and $B$. We also show that a similar analysis can be carried out for particle partition. We discuss the close analogy between the formalism of trial wave functions given by conformal blocks and tensor product states, for which results analogous to ours have appeared recently. Finally, the edge theory and entanglement spectrum of $p_x\pm i{p}_y$ paired superfluids are treated in a similar fashion in the Appendixes.

Figure 1

The edge-state wave functions are obtained (a) by inserting contour integrals in the correlator as in formula (2.7) or equivalently (b) by computing the matrix element (2.8) with an out state $\langle v|$ in radial quantization.

Figure 2

In the thermodynamic limit ($N\to \infty$), the particles fill a circular droplet of radius $R$. For the background potential (1.3) corresponding to the plane, the density is uniform (a), for the sphere it is not uniform after the stereographic projection (b).

Figure 3

(a) Particles in a neutralizing background on the left half-cylinder. We are interested in the correlation function of $a^{†}\left(w_1\right)$ and ${\overline{a}}^{†}\left(\overline{w_2}\right)$ in the right part, in the presence of all the particles. (b) Assuming screening, and in the limit where the density of mobile charges in the left half-cylinder goes to infinity, an operator $a^{†}\left(w_1\right)$ brought to the boundary from the outside is equivalent to an operator $\overline{a}\left(\overline{w_1}\right)$. (c) In this limit, the correlation function of $a^{†}\left(w_1\right)$ and ${\overline{a}}^{†}\left(\overline{w_2}\right)$ in the presence of all the mobile charges in the left is equivalent to the two-point correlation function in the right part, with the boundary condition $\overline{a}(x,y)=a^{†}(x,y)$ at $x=0$.

Figure 4

In the thermodynamic limit, assuming screening, the state (3.13) is equivalent to the conformal boundary state $\left|B\right(R)\rangle$ in the so-called “scaling region.”

Figure 5

We use the completeness relation (4.4) to decompose the conformal correlator $\langle {\prod }_{i}a\left(z_i\right)\rangle$ as a sum of products of one correlator which involve only particles in part $A$ ($\left|z\right|<R$) with another one which involves particles in part $B$ ($\left|z\right|>R$).

Figure 6

(Plots of PP pseudoenergies $\xi$ versus $L_{\widehat{z}}^A$ for $N_A=N/2$ and versus $N_A$ for the $N=12$, $\nu =1/3$ Laughlin state. Here, ${\Delta }^{*}\xi =\xi -N\ln 2$. Main figure: levels in the scaling region versus $L_{\widehat{z}}^A$. The dotted curve is a one-parameter fit of a cubic to seven levels, explained in the text. Inset: lowest pseudoenergy for each $N_A$; the values for $N_A=0$, 12, which are $\xi =N\ln 2$, are omitted. The curve is an inverted parabola, with the same parameter value as the main figure, as explained in the text.

Figure 7

Red: real-space ES of the Laughlin state at $\nu =1/3$ on the sphere, with a cut along the equator (for $N=12$, $N_{A0}=6$). Blue: spectrum of the operator (4.31). Both spectra are plotted in the $\Delta N_A=0$ sector. The inset shows the lowest eigenvalue for the other values of $\Delta N_A$. The parameters $\alpha ,\beta ,\gamma$ are obtained from a least-squares fit that includes all the eigenvalues in the shaded (green) area.

Figure 8

Same as Fig. 7, but with six linearly independent operators in $S_{\mathrm{ES}}$, instead of three, leading to a six-parameter fit on the eigenvalues contained in the shaded (green) area.