Haldane statistics in the finite-size entanglement spectra of 1/$m$ fractional quantum Hall states

We conjecture that the counting of the levels in the orbital entanglement spectra (OES) of finite-size Laughlin fractional quantum Hall (FQH) droplets at filling $\nu =1/m$ is described by the Haldane statistics of particles in a box of finite size. This principle explains the observed deviations of the OES counting from the edge-mode conformal field theory counting and directly provides us with a topological number of FQH states inaccessible in the thermodynamic limit—the boson compactification radius. It also suggests that the entanglement gap in the Coulomb spectrum in the conformal limit protects a universal quantity—the statistics of the state. We support our conjecture with ample numerical checks.

Figure 1

Schematic indicating the excitations above the $\nu =1/2$ ground state that count the number of levels in the OES with $N_A=3$ in $l_{\text{enl}}=7$ orbitals. Haldane statistics implies that each occupied orbital “blocks” its two neighboring orbitals. Thus, only the excited state in the right-hand box contributes to the level counting of the OES at $\Delta L_z=1$, while the configuration arising out of the transition (arrow) in the left-hand box does not.

Figure 2

OES of the bosonic $m=2$ Laughlin state for $N=8$ and (a) $N_A=3$, $l_A=7$, (b) $N_A=3$, $l_A=5$, (c) $N_A=3$, $l_A=4$ and OES of the $m=2$ state with $N=12$, and (d) $N_A=6$ and $l_A=11$ in the conformal limit. The counting of (a)–(c) should be compared to the number of $(1,2)$-admissible configurations listed in Table at each $\Delta L_z$. In all cases, the full counting is predicted exactly by $\mathcal{N}(l_A,N_A,\Delta L_z)$, defined by Eq. (2).

Figure 3

Coulomb entanglement spectrum of $N=8$ bosons at filling $1/2$ in the sector $l_A=7$, $N_A=3$ in the conformal limit. Left-hand side: The full entanglement spectrum with a clear entanglement gap. Right-hand side: The low-“energy” part of the conformal entanglement spectrum. The level counting is identical to the Laughlin entanglement spectrum in Fig. 2a.