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

Abstract

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 $S/p=\gamma \le 1/D$, and $\gamma$ depends on the fractal considered.

5 pages, 2 figures, updated to published version

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• Area law for random graph states
• Entanglement entropy of random partitioning
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