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

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

2009-03-25

arxiv.org/abs/0903.4444v2

Phys. Rev. A 81, 010102(R) (2010)

Physics

Quantum Physics

5 pages, 2 figures, updated to published version

Scientific paper

10.1103/PhysRevA.81.010102

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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