Positivity, negativity, and entanglement

TL;DR

This work analyzes the universal parts of entanglement entropy and logarithmic negativity in 4d CFTs, revealing that their signs are not universally positive and depend on the topology of the entangling surface as well as the central charges. By expressing the universal Rényi entropy in terms of the Willmore energy and Gauss-Codazzi relations, the authors show that for $a>c$ the universal entanglement entropy can become negative for sufficiently high-genus surfaces, with a critical genus $g_c$ marking the transition. They also demonstrate that the ratio ${\widehat{\cal X}}$ defining the relation between negativity and entropy can be negative in the same regime, while for $a\le c$ positivity is preserved in most cases studied. The paper provides explicit results for several free and holographic CFTs, identifies a tight link between topology and central charges, and raises questions about the universality of these sign constraints and their holographic implications.

Abstract

We explore properties of the universal terms in the entanglement entropy and logarithmic negativity in 4d CFTs, aiming to clarify the ways in which they behave like the analogous entanglement measures in quantum mechanics. We show that, unlike entanglement entropy in finite-dimensional systems, the sign of the universal part of entanglement entropy is indeterminate. In particular, if and only if the central charges obey $a>c$, the entanglement across certain classes of entangling surfaces can become arbitrarily negative, depending on the geometry and topology of the surface. The negative contribution is proportional to the product of $a-c$ and the genus of the surface. Similarly, we show that in $a>c$ theories, the logarithmic negativity does not always exceed the entanglement entropy.