Entangling Fractals

TL;DR

This paper calculates the entanglement entropy for fractals using the Heat Kernel method, revealing a novel log-periodic oscillation and connecting fractal spectral dimension with holographic models.

Contribution

It introduces a new approach to compute entanglement entropy on fractals and links spectral dimension with holographic entanglement entropy calculations.

Findings

Discovery of log-periodic oscillations in entropy expressions

Identification of spectral dimension as key in entropy scaling

Agreement between fractal and holographic entropy calculations

Abstract

We use the Heat Kernel method to calculate the Entanglement Entropy for a given entangling region on a fractal. The leading divergent term of the entropy is obtained as a function of the fractal dimension as well as the walk dimension. The power of the UV cut-off parameter is (generally) a fractional number which indeed is a certain combination of these two indices. This exponent is known as the spectral dimension. We show that there is a novel $\log$ periodic oscillatory behavior in the expression of entropy which has root in the complex dimension of the fractal. We finally indicate that the Holographic calculation in a certain hyper-scaling violating bulk geometry yields the same leading term for the entanglement entropy, if one identifies the effective dimension of the hyper-scaling violating theory with the spectral dimension of the fractal. We provide more supports with comparing the behavior of the thermal entropy in terms of the temperature computed for a fractal geometry and a hyper-scaling violating theory as well.