Topological Entanglement in Abelian and Non-Abelian Excitation Eigenstates

Entanglement in topological phases of matter has so far been investigated through the perspective of their ground-state wave functions. In contrast, we demonstrate that the excitations of fractional quantum Hall (FQH) systems also contain information to identify the system’s topological order. Entanglement spectrum of the FQH quasihole (QH) excitations is shown to differentiate between the conformal field theory (CFT) sectors, based on the relative position of the QH with respect to the entanglement cut. For Read-Rezayi model states, as well as Coulomb interaction eigenstates, the counting of the QH entanglement levels in the thermodynamic limit matches exactly the CFT counting, and sector changes occur as non-Abelian quasiholes successively cross the entanglement cut.

Figure 1

FQH sphere with a monopole of $N_{\Phi }$ magnetic flux quanta in the center, partitioned into hemispheres $A$ and $B$, and containing the bosonic Moore-Read state with two QHs. We start with two separated QHs (left) and drag one QH from the north to the south hemisphere (middle). The moving QH is azimuthally delocalized. We end up with a QH twice the charge at the south pole (right).

Figure 2

Conformal-limit ES of the Laughlin model state of $N=12$ bosons with a single QH, localized at one of the poles (a), the equator (c) and in between (b). The location of the QH is given by the dot above each root partition. Insets show high-probability ES levels.

Figure 3

Conformal-limit ES of the Moore-Read model state for $N=20$ bosons with a unit flux added. (a) Abelian vortex $0202\dots 02$. (b) Two fractionalized $e/2$ non-Abelian QHs. Insets show high-probability ES levels.

Figure 4

Conformal-limit ES of the ${\mathbb{Z}}_4$ Read-Rezayi state of $N=28$ bosons and a unit flux added. Abelian vortex (a) is fractionalized and the non-Abelian $e/4$ QHs are moved to the opposite pole one at a time [(b),(c)], causing the sector change. Insets show high-probability ES levels.

Figure 5

Conformal-limit ES of the Coulomb $\nu =1/2$ state of $N=12$ bosons with a single QH, localized using delta impurity potential, at one of the poles $m=0$ (a), the equator $m=12$ (c), and in between $m=6$ (b). The cut is the same as in Fig. 2.

Figure 6

Conformal-limit ES of the Coulomb $\nu =1$ state for $N=16$ bosons and a unit flux added. Delta impurity is located at each pole to pin the non-Abelian QH, as seen in the $e/2$ step in the plot of the excess charge (left-hand inset). The cut is defined by $N_A=8$ and $l_A=8$. Right-hand inset shows the zoom on the lowest ES levels.

Figure 7

Conformal-limit ES of the Laughlin model state of $N=12$ bosons on the sphere with a unit flux removed. The quasielectron is locked to a delta impurity at one of the poles ($m=0$) (a) and displaced halfway toward the equator ($m=6$) (b).