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Entanglement Entropy at Large Central Charge

Thomas Hartman

TL;DR

The paper demonstrates that in 2d CFTs with large central charge and a sparse light spectrum, the leading large-$c$ contribution to the Renyi entropies of N disjoint intervals is universal and governed solely by the Virasoro vacuum block. By formulating and solving a monodromy problem for semiclassical conformal blocks, it shows that entanglement entropies factorize into sums of single-interval entropies in the appropriate channels, with a phase structure mirroring holographic minimal-surface prescriptions. The results reproduce the Ryu–Takayanagi formula for holographic entanglement entropy and its generalization to multiple intervals, providing a microscopic CFT understanding via vacuum exchanges and Liouville theory connections. The approach extends to k-point blocks and N-interval configurations, establishing a deep link between 3d gravity, Liouville theory, and Virasoro symmetry in the semiclassical regime. It also discusses the limitations and potential extensions, including phase transitions and non-vacuum contributions beyond leading order.

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.

Entanglement Entropy at Large Central Charge

TL;DR

The paper demonstrates that in 2d CFTs with large central charge and a sparse light spectrum, the leading large- contribution to the Renyi entropies of N disjoint intervals is universal and governed solely by the Virasoro vacuum block. By formulating and solving a monodromy problem for semiclassical conformal blocks, it shows that entanglement entropies factorize into sums of single-interval entropies in the appropriate channels, with a phase structure mirroring holographic minimal-surface prescriptions. The results reproduce the Ryu–Takayanagi formula for holographic entanglement entropy and its generalization to multiple intervals, providing a microscopic CFT understanding via vacuum exchanges and Liouville theory connections. The approach extends to k-point blocks and N-interval configurations, establishing a deep link between 3d gravity, Liouville theory, and Virasoro symmetry in the semiclassical regime. It also discusses the limitations and potential extensions, including phase transitions and non-vacuum contributions beyond leading order.

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.

Paper Structure

This paper contains 13 sections, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Semiclassical conformal blocks
  3. General operators
  4. Unit operator
  5. Entanglement entropy of two intervals
  6. CFT calculation
  7. Comparison to holography
  8. More than two intervals
  9. Semiclassical $k$-point conformal blocks
  10. Entanglement entropy
  11. Discussion
  12. Calculations of the conformal block
  13. Phases of the $N=n=2$ Renyi entropy

Figures (2)

  • Figure 1: The vacuum contribution to the Renyi entropy (divided by $c/6$), for $n=2,5,10,40$, from bottom to top. In (a) we show the contribution to the $s$-channel, and in (b) both channels. These are computed from (\ref{['sn']}) using the recursion relation in appendix \ref{['app:numerics']} iterated 10 times. The error, estimated by adjusting $c$ and discarding the last term in the recursion, is less than 1% for $z<0.99$. The $\log \epsilon$ UV divergence is dropped.
  • Figure 2: Example of the entanglement entropy of 5 disjoint intervals at large central charge in a particular channel, (a) in CFT and (b) holographically. In (a), trivial monodromy is imposed on cycles in the $z$-plane corresponding to the dashed lines. The solid semicircles in (b) are geodesics in AdS$_3$ with radial direction $r$, and $A$ is the shaded region on the boundary.