Domain Wall Topological Entanglement Entropy

We study the ground-state entanglement of gapped domain walls between topologically ordered systems in two spatial dimensions. We derive a universal correction to the ground-state entanglement entropy, which is equal to the logarithm of the total quantum dimension of a set of superselection sectors localized on the domain wall. This expression is derived from the recently proposed entanglement bootstrap method.

Figure 1

Summary of our assumptions. We assume that $(S_C+S_{BC}-S_B{)}_{\sigma }=0$ (red) and $(S_{BC}+S_{CD}-S_B-S_D{)}_{\sigma }=0$ (green), both in the bulk and on the domain wall, over arbitrarily large regions that can be smoothly deformed from the shown configurations. Here, $(\dots {)}_{\sigma }$ means that the entanglement entropies appearing in the parentheses is computed with respect to the reference state $\sigma$. The subsystems are allowed to be deformed as long as the boundaries between $B$ and $D$ do not cross the domain wall.

Figure 2

(a) Subsystems involved in the definition of the domain wall topological entanglement entropy. (b) Three ways to detect a pointlike excitation (red dot) on the domain wall. Measurement processes are considered in the blue regions. The parton sectors can be detected from a measurement process strictly localized on $N$-shaped (left) and $U$-shaped (middle) subsystems. In comparison, the set of superselection sectors of pointlike excitations in the vicinity of the gapped domain wall can be detected from a measurement on $O$-shaped (right) subsystems.

Figure 3

(a) Subsystems relevant to the domain wall topological entanglement entropy $S_{\mathrm{topo},N}$. (b) A partition of $N$ into $ABC$. $BCD$ is a disk.

Figure 4

(a) Partition of $N=ABC$. (b) Further partition $B=B_1{B}_2$ and subsystems relevant to merging.

Figure 5

The partition used in the proof of Eq. (2). $D_1{D}_2=D$.