Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches (1410.1392v2)

Abstract: We consider the entanglement entropy in 2d conformal field theory in a class of excited states produced by the insertion of a heavy local operator. These include both high-energy eigenstates of the Hamiltonian and time-dependent local quenches. We compute the universal contribution from the stress tensor to the single interval Renyi entropies and entanglement entropy, and conjecture that this dominates the answer in theories with a large central charge and a sparse spectrum of low-dimension operators. The resulting entanglement entropies agree precisely with holographic calculations in three-dimensional gravity. High-energy eigenstates are dual to microstates of the BTZ black hole, so the corresponding holographic calculation is a geodesic length in the black hole geometry; agreement between these two answers demonstrates that entanglement entropy thermalizes in individual microstates of holographic CFTs. For local quenches, the dual geometry is a highly boosted black hole or conical defect. On the CFT side, the rise in entanglement entropy after a quench is directly related to the monodromy of a Virasoro conformal block.

• The paper calculates entanglement entropy via heavy operator insertions, highlighting the universal contribution of the stress tensor in large-central-charge CFTs.
• The paper analyzes time-dependent local quenches, showing their effect on entanglement growth through conformal block dynamics and Virasoro monodromy.
• The paper finds precise numerical agreement between 2D CFT computations and holographic gravitational models, reinforcing the connection with BTZ black hole microstates.

Holographic Entanglement Entropy in 2D CFT: An Analysis of Heavy States and Local Quenches

The paper under examination elucidates the computation of entanglement entropy in two-dimensional conformal field theory (2d CFT) by inserting heavy local operators to create excited states. These states may either represent high-energy eigenstates linked to the Hamiltonian or result from time-dependent local quenches. The research primarily focuses on calculating the universal contribution of the stress tensor to the Renyi entropies and entanglement entropy for a single interval. A conjecture is presented that stresses the domination of this universal contribution in CFTs possessing a large central charge and a sparse low-dimension operator spectrum. The paper intriguingly correlates the calculated entanglement entropies with those derived from holographic gravitational models, specifically in three-dimensional gravity scenarios.

Main Themes and Contributions

1. High-Energy Eigenstates and Holography: The high-energy eigenstates discussed are conceived as duals to BTZ black hole microstates, creating a compelling connection with black hole physics. The computed holographic entanglement entropy in these eigenstates, which stems from the geodesic length in black hole geometries, reflects a scenario where entanglement entropy thermalizes microstate-wise in holographic CFTs.
2. Time-dependent Local Quenches: For scenarios involving local quenches, the holographically dual geometries transform into highly boosted black holes or conical defects. The paper details how the entanglement entropy increase, post-quench, connects to the monodromy of a Virasoro conformal block, offering a deeper understanding of quantum entanglement dynamics in perturbed systems.
3. Twist Operators and Conformal Blocks: The analysis utilizes the replica trick to express Renyi entropies through path integrals involving twist operators. Particularly, it expands upon a conformal block decomposition to deliberate contributions from identity blocks, advocating their prominence in sparse CFTs with a large central charge.
4. Holographic Calculations and Results Agreement: Concrete numerical matches emerge between the theoretical CFT calculations and holographic counterparts, establishing the consistency and reliability of holographically derived entanglement entropy even in dynamic and excited states.

Implications and Speculations

• Theoretical: The findings extend the holographic principle's applicability context within CFT, reinforcing the bridge between generalized entangled states and gravitational duals. The precise agreement with holographic results posits these calculations as a testament to holographic principles, offering a more expansive framework for understanding entanglement's role in quantum gravity formulations.
• Practical: These insights could significantly inform computational techniques for evaluating similar entropy metrics in more complex quantum systems, possibly extending to non-conformal or higher-dimensional settings. The methodologies could serve as templates for quantum simulations aiming to explore entangled states' behaviors in analogous holographic models.

Future Directions

1. Exploration Beyond Large N Limit: The paper could be extended to examine how the conclusions drawn under the large $N$ approximation might hold or differ when considering finite $N$, exploring the efficacy of identity block dominance in less sparse CFTs.
2. Non-Vacuum State Applications: Investigating non-vacuum states and their resulting geometries could reveal additional insights into how all microstates beyond trivial vacua influence the entanglement structure in more intricate CFTs.
3. Entanglement Entropy in Diverse Geometries: Introducing manifold variations in the bulk, such as those involving different topological or boundary conditions, could facilitate understanding the impact of geometric modifications on entangled systems.

In summary, this research paper successfully bridges entanglement entropy concepts in 2D CFT with holographic principles, providing rigorous numerical consistency and rich theoretical insights into quantum states dynamics. These advancements present substantial opportunities for further exploration in the domain of conformal field theories and quantum gravity integration.