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

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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 Hausdorff dimension D, we show that the entanglement between the two parts scales as S/p=γ≤1/D, and γ depends on the fractal considered.