On entanglement c-functions in confining gauge field theories

TL;DR

This paper studies entanglement-based c-functions in holographic RG flows, revealing non-monotonic behavior in higher dimensions due to geometric transitions and proposing modifications to restore monotonicity.

Contribution

It identifies the cause of non-monotonicity in entanglement c-functions in higher dimensions and proposes a modified construction for flows from four to three dimensions.

Findings

Standard c-functions are non-monotonic in $d \,\geq\, 4$ due to geometric transitions.

A modified c-function restores monotonicity in the infrared for certain compactifications.

A conjectured bound on the cylinder entanglement c-function holds across studied examples.

Abstract

Entanglement entropy has proven to be a powerful tool for probing renormalization group (RG) flows in quantum field theories, with c-functions derived from it serving as candidate measures of the effective number of degrees of freedom. While the monotonicity of such c-functions is well established in many settings, notable exceptions occur in theories with a mass scale. In this work, we investigate entanglement c-functions in the context of holographic RG flows, with a particular focus on flows across dimensions induced by circle compactifications. We argue that in spacetime dimensions $d \geq 4$, standard constructions of c-functions, which rely on higher derivatives of the entanglement entropy of either a ball or a cylinder, generically lead to non-monotonic behavior. Working with known dual geometries, we argue that the non-monotonicity stems not from any pathology or curvature singularity, but from a transition in the holographic Ryu--Takayanagi surface. In compactifications from four to three dimensions, we propose a modified construction that restores monotonicity in the infrared, although a fully monotonic ultraviolet extension remains elusive. Furthermore, motivated by entanglement entropy inequalities, we conjecture a bound on the cylinder entanglement c-function, which holds in all our examples.