On entanglement c-functions in confining gauge field theories

TL;DR

This paper investigates entanglement-based c-functions in holographic RG flows, emphasizing flows across dimensions induced by circle compactifications. It shows that in $d\ge 4$, standard constructions from ball or cylinder entanglement typically yield non-monotonic RG trajectories, with the non-monotonicity traced to transitions of the Ryu--Takayanagi surface rather than geometric pathologies. The authors propose an IR-adapted approach for 4-to-3 flows that can restore monotonicity in the IR and explore conjectured bounds on cylinder c-functions tied to entanglement inequalities; they also compare entanglement c-functions with holographic flow c-functions which often remain monotonic in top-down models. Through both field-theory (Four-dimensional ${\cal N}=2$ linear quivers) and top-down holographic constructions (e.g., KS and GPPZ backgrounds), the work reveals that RT-surface transitions underpin non-monotonicity and discusses circumstances under which alternative c-functions provide monotone, scheme-independent measures of degrees of freedom. Overall, the study highlights the challenge of defining a fully monotone entanglement c-function in confining holographic theories and motivates further development of covariant, monotone probes of RG flow.

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.

Figures (15)

• Figure 1: Long quiver of length $P-1$ with gauge nodes $N_i$ and flavor nodes $F_i$. The quiver is balanced and conformal if $F_i = 2 N_i - N_{i-1}-N_{i+1}$.
• Figure 2: Illustration of two of the possible types of embeddings that solve \ref{['eq:eom3']}. Cases (a) and (b) follow from setting \ref{['eq:bc1']} and \ref{['eq:bc2']} as a boundary condition, respectively. Figure adapted from Jokela:2020wgs.
• Figure 3: Numerical results for the small radius (blue) and large radius (orange) embeddings $\varrho(\zeta)$. We will keep this color coding in what follows. The dashed line is not a real boundary, but the surface where the $S^1$ of the background shrinks to zero size.
• Figure 4: Renormalized entanglement entropy as a function of the radius with the zoomed in version focusing on the swallowtail on the right panel.
• Figure 5: (Left) Proposed measure of the effective number of degrees of freedom from the lower dimensional perspective, \ref{['eq:subtracted_F_cyl']}, for different choices of $R_{\text{\tiny{ref}}}r_{\text{c}} = 0.01$, $0.05$, and $0.25$. The different curves cross the horizontal axes precisely at these same $R=R_{\text{\tiny{ref}}}$. The "desired" properties for a c-function are recovered as $R_{\text{\tiny{ref}}}$ becomes small. The discontinuity, depicted by the vertical black straight line, corresponds to the phase transition of the entanglement entropy, see Fig. \ref{['fig:EE']}. (Right) Cylinder c-function as given in \ref{['eq:Ccyl']} is depicted as a function of the base radius $R$.