Entanglement as a Probe of Confinement (0709.2140v2)

Abstract: We investigate the entanglement entropy in gravity duals of confining large $N_c$ gauge theories using the proposal of arXiv:hep-th/0603001, arXiv:hep-th/0605073. Dividing one of the directions of space into a line segment of length $l$ and its complement, the entanglement entropy between the two subspaces is given by the classical action of the minimal bulk hypersurface which approaches the endpoints of the line segment at the boundary. We find that in confining backgrounds there are generally two such surfaces. One consists of two disconnected components localized at the endpoints of the line segment. The other contains a tube connecting the two components. The disconnected surface dominates the entropy for $l$ above a certain critical value $l_{\rm crit}$ while the connected one dominates below that value. The change of behavior at $l=l_{\rm crit}$ is reminiscent of the finite temperature deconfinement transition: for $l < l_{\rm crit}$ the entropy scales as $N_c^2$, while for $l > l_{\rm crit}$ as $N_c^0$. We argue that a similar transition should occur in any field theory with a Hagedorn spectrum of non-interacting bound states. The requirement that the entanglement entropy has a phase transition may be useful in constraining gravity duals of confining theories.

• The paper demonstrates that holographic entanglement entropy distinguishes between connected and disconnected minimal surface phases in confining gauge theories.
• It employs minimal hypersurface computations to reveal a critical length where the entropy transition mirrors a deconfinement to confinement shift.
• The findings constrain holographic models by establishing entropic behaviors as universal markers for confinement in large-Nc quantum field theories.

Entanglement Entropy as a Probe of Confinement in Holographic Models

The paper presented in the paper titled "Entanglement as a Probe of Confinement" investigates the behavior of entanglement entropy within gravity duals of confining large $N_c$ gauge theories. The authors explore how the holographic entanglement entropy can reveal features of the confining phase of quantum field theories (QFTs) with a large number of colors $N_c$.

Key Findings and Methodology

The paper uses the holographic entanglement entropy proposal formulated earlier, which relates the entropy between subregions in QFTs to the classical action of minimal bulk hypersurfaces. By dividing spatial dimensions into a line segment of length $l$ and considering its complement, distinct behaviors for entanglement entropy are uncovered in confining backgrounds:

• Disconnected and Connected Minimal Surfaces: The paper identifies two kinds of minimal surfaces within these gravity backgrounds. One surface is disconnected, localized at the endpoints of the dividing segment, and the other is connected, forming a tube that links the two components. Notably, a transition occurs at a critical length $l_{\rm crit}$, above which the disconnected surface dictates the entropy, whereas below this threshold the connected surface is dominant.
• Phase Transition and Scaling: The transition at $l_{\rm crit}$ resembles the finite temperature deconfinement transition. For $l < l_{\rm crit}$, the entropy scales with $N_c^2$, indicative of the degrees of freedom in a deconfined phase, while for $l > l_{\rm crit}$, the scaling shifts to $N_c^0$.

The paper extends its analysis to various specific gravitational backgrounds, such as geometries containing D4 and D3-branes wrapped on circles and the warped deformed conifold (known from the Klebanov-Strassler model). By comparing connected and disconnected solutions across these backgrounds, the authors illustrate the universal presence of such phase transitions in confining field theories with holographic duals.

Implications

This research implies significant constraints on the holographic modeling of confinement: any consistent dual of a confining theory should exhibit a transition in the entanglement entropy similar to the one encountered in their paper. The robustness across different models suggests these entropic behaviors are a haLLMark feature of confinement in holography.

The work connects these holographic results with large $N_c$ asymptotically free gauge theories, proposing analogous behavior due to the Hagedorn spectrum of states inherent to these theories. Hence, the research enriches the understanding of both holographic duals and the field theory regime, repurposing entanglement profiles as a probe into the confining phases of strongly coupled gauge theories.

Future Directions

The insights from this paper provide a promising avenue for further research. Future developments could examine entanglement entropy transitions in more diverse holographic backgrounds or explore further the $1/N_c$ corrections in these setups. Moreover, exploring entanglement behavior in lattice gauge theories might offer non-holographic verification or insights into finite $N_c$ effects, refining our understanding of confinement physics across frameworks.

In summary, the paper delivers significant contributions to the comprehension of confinement in gauge/gravity duality through the lens of entanglement entropy, further solidifying holography as a powerful tool for probing strong coupling dynamics in QFTs.