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arXiv · 0903.4444

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

Alioscia Hamma·Daniel A. Lidar·Simone Severini
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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 $S/p=\gamma\leq1/D$, and $\gamma$ depends on the fractal considered.

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Alioscia Hamma, Daniel A. Lidar, Simone Severini. 2009-03-25. Entanglement and area law with a fractal boundary in a topologically ordered phase. https://doi.org/10.1103/physreva.81.010102

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