Entanglement and area law with a fractal boundary in a topologically ordered phase

Quantum systems with short-range interactions are known to respect an area law for the entanglement entropy: The von Neumann entropy $S$ associated to a bipartition scales with the boundary $p$ between the two parts. Here we study the case in which the boundary is a fractal. We consider the topologically ordered phase of the toric code with a magnetic field. When the field vanishes it is possible to analytically compute the entanglement entropy for both regular and fractal bipartitions ($A,B$) of the system and this yields an upper bound for the entire topological phase. When the $A-B$ boundary is regular we have $S/p=1$ for large $p$. When the boundary is a fractal of the Hausdorff dimension $D$, we show that the entanglement between the two parts scales as $S/p=\gamma ⩽1/D$, and $\gamma$ depends on the fractal considered.

Figure 1

The drawings show different bipartitions of the system. The subsystem $\mathcal{A}$ consists of all the spins marked by the (black) squares. The entanglement is given by the number of plaquette operators acting on both subsystems, marked by (red) dots. For a regular figure (left), this number coincides with the perimeter $p$, which is the number of spins along the boundary (in bold, yellow). Every time there is an inward angle there is one such operator for three units of length. The well (middle) contains two inward angles. A hole (right) of size one accounts for four units of length and contains only one star operator.

Figure 2

Top, left to right: Sierpinski carpet ${\mathcal{S}}_3$, Greek cross ${\mathcal{G}}_3$, type-2 quadratic von Koch curve (Minkowski sausage) ${\mathcal{I}}_3$, and T-square ${\mathcal{E}}_4$. Bottom, left to right: Moore polygons ${\mathcal{M}}_3$, Vicsek fractal ${\mathcal{V}}_3$, half perimeter of the Koch polygon ${\mathcal{K}}_5$, and $4\times 4$ chessboard ${\mathcal{C}}_4$.