Entanglement entropy, conformal invariance and extrinsic geometry (0802.3117v4)

Abstract: We use the conformal invariance and the holographic correspondence to fully specify the dependence of entanglement entropy on the extrinsic geometry of the 2d surface $\Sigma$ that separates two subsystems of quantum strongly coupled ${\mathcal{N}}=4$ SU(N) superconformal gauge theory. We extend this result and calculate entanglement entropy of a generic 4d conformal field theory. As a byproduct, we obtain a closed-form expression for the entanglement entropy in flat space-time when $\Sigma$ is sphere $S_2$ and when $\Sigma$ is two-dimensional cylinder. The contribution of the type A conformal anomaly to entanglement entropy is always determined by topology of surface $\Sigma$ while the dependence of the entropy on the extrinsic geometry of $\Sigma$ is due to the type B conformal anomaly.

• The paper derives a comprehensive formula for entanglement entropy that integrates area terms with contributions from both conformal and geometric anomalies in 4D CFTs.
• It utilizes holographic duality, applying AdS/CFT correspondence to analyze quantum correlations in strongly coupled N=4 SU(N) gauge theories.
• It demonstrates that extrinsic curvature and topology critically affect entropy calculations, offering new insights for further quantum gravity research.

Entanglement Entropy, Conformal Invariance, and Extrinsic Geometry

The paper by Sergey N. Solodukhin addresses the relationship between entanglement entropy, conformal invariance, and the extrinsic geometry of the separating surface in quantum field theories, focusing particularly on strongly coupled N = 4 SU(N) superconformal gauge theories. It provides an extended analysis into a generic four-dimensional conformal field theory (CFT).

Entanglement Entropy and Geometry

Entanglement entropy serves as a quantitative measure of quantum correlations between two subsystems demarcated by a surface, noted as Σ. Such entropy, fundamentally influenced by the geometry of the surface, plays a crucial role in black hole physics and quantum field theories. The holographic interpretation reveals a geometric approach for computing the entropy of strongly coupled CFTs.

For these CFTs, the paper presents a derivation of entanglement entropy—referred here as $S_{SC}$—and its dependency on both topology and extrinsic geometry. The form of this entropy expression is:

$$S_{SC} = \frac{A(\Sigma)}{4\pi\epsilon^2} + \frac{N^2}{24\pi} \int_{\Sigma} (3R_{aa} - 2R - 2k_{aa}) \ln \epsilon + s_{SC}(g),$$

which integrates an area term, a contribution from conformal anomalies, and a finite part, $s_{SC}(g)$.

Conformal Anomalies and Extrinsic Curvature

The analysis differentiates the impact of type A and type B conformal anomalies on entanglement entropy. The type A anomaly contribution is invariant with the surface topology. However, the type B anomaly introduces dependency on the extrinsic curvature of Σ. Solodukhin argues that this dependence, previously underexplored, is critical. Particularly, in curved space-time, both intrinsic and extrinsic geometric features contribute to the entropy.

Holographic Correspondence

Through the holographic duality, the paper aligns results with the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, particularly for theories like the N = 4 SU(N) super Yang-Mills theory. The holographic duality supports the results with additional insight into how entanglement spans minimal surfaces in the AdS space. Detailed calculations for surfaces such as the two-dimensional cylinder and the sphere $S^2$ validate this correspondence.

Numerical Results and Theoretical Implications

By calculating and comparing entanglement entropy for different surface geometries, the paper discerns that distinct conformal anomalies are isolated effectively using specific geometric configurations. For example, in flat space-time, the entanglement entropy of a generic CFT for a cylindrical surface is highlighted by:

$$S_{A,B}^{cylinder} = \frac{A(\Sigma)}{4\pi \epsilon^2} + \frac{\pi^2B}{8a}\ln \frac{L}{a},$$

whereas for $S^2$, it is:

$$S_{A,B}^{sphere} = \frac{A(\Sigma)}{4\pi \epsilon^2} + A\pi \ln \frac{2\pi a}{\epsilon}.$$

The results underscore significant relations between geometric properties and theory-intrinsic anomalies, crucial for understanding both theoretical and practical scenarios in conformal field theories.

Future Directions

The examination of entanglement entropy through the lens of extrinsic geometry offers enlightening perspectives for further exploration within quantum field theories, especially as they pertain to black hole information and quantum gravity. Future research may explore computational methodologies or extend the analyses to a broader class of quantum systems with varied interaction profiles.

In summary, this paper advances the theoretical framework when examining the entanglement entropy associated with diverse geometrical and topological configurations in four-dimensional CFTs. The contributions provide a robust foundation for further explorations of both quantum field theoretical properties and holographic principles.