Disentangling Entanglement Spectra of Fractional Quantum Hall States on Torus Geometries

We analyze the entanglement spectrum of Laughlin states on the torus and show that it is arranged in towers, each of which is generated by modes of two spatially separated chiral edges. This structure is present for all torus circumferences, which allows for a microscopic identification of the prominent features of the spectrum by perturbing around the thin-torus limit.

Figure 1

Torus setup for block entanglement computations. The lowest Landau level is spanned by orbitals which in Landau gauge are centered along the circles shown. The arrows indicate the chiralities of the virtual “edges” created by the block partitioning.

Figure 2

Entanglement spectrum for $N_s=36$ and $L_1=10$. Left panels show the symmetric cut and right panels show one of the asymmetric cuts. The origin of $K_A$ is chosen to match the Tao-Thouless state. The blue squares represent numerically obtained data. The assigned edge modes are labeled by black dots while the combinations of those edges are marked by red crosses. The script letters are microscopic identifiers for the two edges combining to form each tower (see text). The striking correspondence of the red crosses with numerical data shows that our algorithm based on the two-edge picture allows the reconstruction of the entire entanglement spectrum using only the positions of the black dots. The inset shows a CFT tower formed by two ideal chiral edges, the states labeled with their degeneracies.

Figure 3

(a) The chiral edge levels (type $\mathcal{A}$) identified from ES, as a function of torus thickness $L_1$. The rectangle contains the single-edge ES levels in spherical geometry [15], here scaled and shifted for best comparison with the torus results around $L_1\sim 14$. (b) “Aspect ratio” of the diamond formed by the four lowest ES levels of the $\mathcal{A}\mathrm{\text{−}}\mathcal{A}$ tower. The value 2 means a perfect diamond shape.

Figure 4

Comparison of the entanglement spectrum, and overlaps, between the Laughlin wave function and the Coulomb ground state for $L_1=6,\dots ,14$ at fixed $N_s=36$. Only the central part of the most prominent tower for the symmetric cut is displayed. We observe that the entanglement spectra of the two wave functions become very similar for sufficiently large $L_1$. The appearance of “generic levels” beyond the two-edge CFT picture in the Coulomb state is indicated by the shaded regions, leading to a tentative notion of “entanglement gap” [4].