Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States

TL;DR

It is proposed that the low-lying entanglement spectrum can be used as a "fingerprint" to identify topological order and is compared with a generic 5/2 state obtained by finite-size diagonalization of the second-Landau-level-projected Coulomb interactions.

Abstract

We study the "entanglement spectrum" (a presentation of the Schmidt decomposition analogous to a set of "energy levels") of a many-body state, and compare the Moore-Read model wavefunction for the $ν$ = 5/2 fractional quantum Hall state with a generic 5/2 state obtained by finite-size diagonalization of the second-Landau-level-projected Coulomb interactions. Their spectra share a common "gapless" structure, related to conformal field theory. In the model state, these are the \textit{only} levels, while in the "generic" case, they are separated from the rest of the spectrum by a clear "entanglement gap", which appears to remain finite in the thermodynamic limit. We propose that the low-lying entanglement spectrum can be used as a "fingerprint" to identify topological order.

Figures (3)

• Figure 1: The complete entanglement spectra of the $N_e=16$ and $N_{orb}=30$ Moore-Read state (only the relative values of $\xi$ and $L_z^A$ are meaningful).
• Figure 2: The low-lying entanglement spectra of the $N_e=16$ and $N_{orb}=30$ ground state of the Coulomb interaction projected into the second Landau level (there are levels beyond the regions shown here, but they are not of interest to us). The insets show the low-lying parts of the spectra of the Moore-Read state, for comparison [see Figure (\ref{['mrfig']})]. Note that the structure of the low-lying spectrum is essentially identical to that of the ideal Moore-Read state.
• Figure 3: Entanglement gap as a function of $1/N$. $\delta_0$ is the gap at $\Delta L=0$, i.e., the distance from the single CFT level at $\Delta L=0$ to the bottom of the generic (non-CFT) levels at $\Delta L=0$. At $\Delta L=1,2$, the gap $\delta_{1,2}$ is defined as the distance from the average of the CFT levels to the bottom of the generic levels. See Table \ref{['table']} for the details of various partitionings.