On the Holographic Entanglement Entropy for Non-smooth Entangling Curves in AdS(4)

TL;DR

This paper extends holographic entanglement entropy calculations in AdS(4) to non-smooth entangling curves with singular points, revealing a logarithmic term dependent on local geometry and minimal surface uniqueness.

Contribution

It introduces a method to analyze entanglement entropy for curves with complex singularities using elliptic solutions, highlighting the importance of local geometric characteristics.

Findings

Logarithmic terms appear in entanglement entropy for non-smooth curves.

The logarithmic term depends solely on local geometry of singular points.

Smoothness of the entropy's dependence on geometric characteristics is not always guaranteed.

Abstract

We extend the calculations of holographic entanglement entropy in AdS(4) for entangling curves with singular non-smooth points that generalize cusps. Our calculations are based on minimal surfaces that correspond to elliptic solutions of the corresponding Pohlmeyer reduced system. For these minimal surfaces, the entangling curve contains singular points that are not cusps, but the joint point of two logarithmic spirals one being the rotation of the other by a given angle. It turns out that, similarly to the case of cusps, the entanglement entropy contains a logarithmic term, which is absent when the entangling curve is smooth. The latter depends solely on the geometry of the singular points and not on the global characteristics of the entangling curve. The results suggest that a careful definition of the geometric characteristic of such a singular point that determines the logarithmic term is required, which does not always coincide with the definition of the angle. Furthermore, it is shown that the smoothness of the dependence of the logarithmic terms on this characteristic is not in general guaranteed, depending on the uniqueness of the minimal surface for the given entangling curve.