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

TL;DR

This paper investigates how entanglement entropy scales with fractal boundaries in topologically ordered quantum phases, revealing a relationship between the entropy and the fractal's Hausdorff dimension.

Contribution

It analytically derives the entanglement entropy scaling for fractal boundaries in the toric code model, extending the area law to fractal geometries.

Findings

Entanglement entropy scales as S/p ≤ 1/D for fractal boundaries with Hausdorff dimension D.

For regular boundaries, the entropy-to-boundary ratio approaches 1.

The scaling factor γ depends on the specific fractal boundary considered.

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.