Entanglement Entropy at Large Central Charge (1303.6955v1)

Abstract: Two-dimensional conformal field theories with a large central charge and a small number of low-dimension operators are studied using the conformal block expansion. A universal formula is derived for the Renyi entropies of N disjoint intervals in the ground state, valid to all orders in a series expansion. This is possible because the full perturbative answer in this regime comes from the exchange of the stress tensor and other descendants of the vacuum state. Therefore, the Renyi entropy is related to the Virasoro vacuum block at large central charge. The entanglement entropy, computed from the Renyi entropy by an analytic continuation, decouples into a sum of single-interval entanglements. This field theory result agrees with the Ryu-Takayanagi formula for the holographic entanglement entropy of a 2d CFT, applied to any number of intervals, and thus can be interpreted as a microscopic calculation of the area of minimal surfaces in 3d gravity.

• The paper provides a universal expression for Renyi entropies in 2D CFTs with large central charge via a detailed analysis of the Virasoro vacuum block.
• It employs the replica trick and recursive expansion to derive semiclassical conformal blocks with high precision, validating agreement with holographic predictions.
• The study bridges field theory with holographic geometry, reinforcing the equivalence with the Ryu-Takayanagi formula through explicit numerical and analytical results.

Entanglement Entropy in 2D Conformal Field Theories at Large Central Charge

The paper by Thomas Hartman focuses on the computation of entanglement entropy in two-dimensional conformal field theories (CFTs) with a large central charge, using a comprehensive analysis of the conformal block expansion. The investigation revolves around providing a universal expression for the Renyi entropies of multiple disjoint intervals in the ground state that holds to all orders in perturbative expansion due to contributions from the stress tensor and other descendants of the vacuum state. The relationship between the Renyi entropy and the Virasoro vacuum block plays a central role, providing a link with the holographic Ryu-Takayanagi formula for entanglement entropy.

Theological and Methodological Insights

The manuscript explores the entanglement entropy of a subsystem formed by disjoint intervals, illustrating its dependence on the complete operator content of the CFT. In the large-central-charge domain, where theories have reduced numbers of light operators, the entanglement entropy exhibits universal characteristics—determined predominantly by conformal invariance. The paper is motivated further by implications within the AdS/CFT correspondence, emphasizing theories with holographic duals characterized by a simple semiclassical gravitational description.

The authors employ the replica trick to compute Renyi entropies, defining them in terms of $2N$-point correlation functions, analogously leading to entanglement entropy via analytic continuation. The provision of a universal formula for entanglement entropy underscores its alignment with the Ryu-Takayanagi proposal, establishing an equivalence in computation with regard to the area of minimal surfaces in three-dimensional gravity.

Numerical and Analytical Results

A notable contribution is the derivation and computational representation of the semiclassical conformal block in cases of large central charge, particularly focusing on maintaining high precision through a recursive expansion approach. This procedure enables the verification of agreement between leading-order CFT results and holographic entanglement entropy, overcoming potential challenges of non-perturbative effects or higher-dimensional extensions.

Additionally, through the employment of the monodromy method, the paper analytically derives that entanglement entropy decouples into a summation over single-interval contributions. Numerical calculations provide explicit expansions and match preceding holographic predictions, thus reinforcing the assumption that, in any channel, the dominant contribution stems from the vacuum block.

Implications and Future Prospects

The implications posit a foundational understanding of entanglement entropy in 2D CFTs with large central charge, equating it in a universal manner with geometric definitions in holographic theories. Furthermore, the work opens avenues toward interpreting gravitational sectors of AdS/CFT in terms of stress tensor representations in $d$-dimensional CFTs, essentially controlling the thermodynamic sector of the theory.

Possible future developments could explore deeper relations in higher-dimensional CFTs where the vacuum representation changes nature, or extend to CFTs with higher spin symmetry, which can provide alternative insights into the field-theoretic and dual gravitational descriptions. Finally, validating and generalizing such universal properties to varying boundary conditions or expansions in complexity remains a rich field for exploration. Such insights could contribute significantly to our understanding of the emergent phenomena in quantum gravity and the holographic principle.

This work is a substantial step forward in linking field-theoretical methods with geometrical intuitions in holographic settings, reinforcing the interconnections and providing a refined toolkit for addressing these theoretical frameworks.