Entanglement entropy on finitely ramified graphs

TL;DR

This paper analyzes how entanglement entropy behaves on finitely ramified, self-similar lattice graphs, revealing dependence on the decimation factor and spectral dimension, with implications for holographic theories.

Contribution

It introduces a detailed computation of entanglement entropy on self-similar graphs, highlighting the effects of spectral dimension and higher order oscillatory corrections.

Findings

Entanglement entropy increases with decimation factor, approaching a constant asymptotically.

Small decimation factors lead to reduced entropy due to spectral gaps.

Higher order corrections exhibit log-periodic oscillations.

Abstract

We compute the entanglement entropy in a composite system separated by a finitely ramified boundary with the structure of a self-similar lattice graph. We derive the entropy as a function of the decimation factor which determines the spectral dimension, the latter being generically different from the topological dimension. For large decimations, the graph becomes increasingly dense, yielding a gain in the entanglement entropy which, in the asymptotically smooth limit, approaches a constant value. Conversely, a small decimation factor decreases the entanglement entropy due to a large number of spectral gaps which regulate the amount of information crossing the boundary. In line with earlier studies, we also comment on similarities with certain holographic formulations. Finally, we calculate the higher order corrections in the entanglement entropy which possess a log-periodic oscillatory behavior.