Confinement, Phase Transitions and non-Locality in the Entanglement Entropy

TL;DR

The paper investigates the link between confinement and a first-order phase transition in holographic entanglement entropy (EE), deriving explicit IR and UV conditions that govern the existence of the EE transition and testing them across several backgrounds including AdS, D$p$-brane on a circle, hard/soft walls, and Klebanov–Strassler (KS).A central finding is that non-local UV behavior can erase the EE phase transition, but introducing a UV cutoff or performing a UV completion (via the KS baryonic branch or carefully localized sources) can restore the transition and thereby reveal a holographic signature of locality.The work highlights the role of short configurations near the UV boundary as a mechanism to reintroduce the EE transition in non-local theories, and it establishes a conceptual bridge between EE, Wilson loops, and the UV structure of the dual QFT, with implications for identifying confining, local 4D QFTs in holographic models.Overall, EE serves as a diagnostic tool for both confinement and locality, and the paper provides concrete recipes to diagnose, regulate, and complete non-local holographic models to recover physically meaningful EE phase transitions.

Abstract

In this paper we study the conjectural relation between confinement in a quantum field theory and the presence of a phase transition in its corresponding entanglement entropy. We determine the sufficient conditions for the latter and compare to the conditions for having a confining Wilson line. We demonstrate the relation in several examples. Superficially, it may seem that certain confining field theories with a non-local high energy behaviour, like the dual of D5 branes wrapping a two-cycle, do not admit the corresponding phase transition. However, upon closer inspection we find that, through the introduction of a regulating UV-cutoff, new eight-surface configurations appear, that satisfy the correct concavity condition and recover the phase transition in the entanglement entropy. We show that a local-UV-completion to the confining non-local theories has a similar effect to that of the aforementioned cutoff.

Figures (26)

• Figure 1: The phase diagram for the entanglement entropy in confining theories. On the left, the length of the connected solution as a function of the minimal radial position in the bulk $L({\rho_{0}})$, which is a non-monotonic function in confining theories. On the right, the entanglement entropy of the strip as a function of its length. The solid blue line represent the connected solution while the dashed red line is the disconnected solution. At the point $L=L_c$ there is a first order phase transition between the two solutions. This type of first-order phase transition behavior is called the "butterfly shape" in the bibliography.
• Figure 2: On the left, the length of the Wilson loop as a function of the minimal radial position in the bulk, which is a monotonically decreasing function. On the right, the energy of the Wilson loop as a function of its length. At long distances the energy is linear in length, which correspond to linear confinement.
• Figure 3: The case of $AdS_5\times S^5$ --- Here we plot $L(\rho_0)$ and $S(L)$.
• Figure 4: The function $L(\rho_0)$ and $S(L)$ in the near extremal D$p$ brane backgrounds for $p=3,4,5,6$ moving down the page. The location of the horizon was set to ${\rho_{\Lambda}}=1$ in the figures. The dashed red line is the disconnected solution. The D$3$ and D$4$ branes shows a phase transition behavior while in the D$5$ and D$6$ branes there is no phase transition.
• Figure 5: The function $L(\rho_0)$ and $S(L)$ in the Hard (top row) and Soft Wall (bottom row) models. The location of the hard wall was set to ${\rho_{\Lambda}}=1$ in the figures. The dashed red line is the disconnected solution and the dashed blue line represents the continuation of the $AdS$ solution beyond the hard wall.