Large-c BCFT Entanglement Entropy with Deformed Boundaries from Emergent JT Gravity

This paper demonstrates that at large central charge, the von Neumann entropy of subregions in a two-dimensional boundary conformal field theory with an infinitesimally deformed boundary is exactly reproduced by the island entropy of the corresponding interval in an undeformed BCFT coupled to a two-dimensional Jackiw-Teitelboim gravity system, where the boundary deformation determines the dilaton boundary condition.

The Gist

The Big Picture: A Magic Trick with Boundaries

Imagine you have a piece of fabric (representing a quantum system) that is sitting on a table. The edge of this fabric is its boundary. Usually, in physics, we assume this edge is perfectly straight and doesn't move.

This paper asks a simple question: What happens if we gently wiggle or deform that edge?

The authors discovered something surprising. When you wiggle the edge of a specific type of quantum fabric (a 2D Boundary Conformal Field Theory, or BCFT), the math describing that wiggle looks exactly the same as the math describing a tiny patch of gravity appearing underneath the fabric.

It's as if you didn't just bend the edge of a tablecloth; you secretly turned the tablecloth into a trampoline that bends space-time.

The Two Worlds Being Compared

To prove this, the authors compared two different ways of looking at the same problem:

1. The "Flat" World (The BCFT)

Imagine a flat, 2D world (like a sheet of paper) with a straight edge.

• The Action: You take a ruler and push the edge of the paper slightly inward or outward.
• The Measurement: You measure how "entangled" (how connected) different parts of the paper are. In quantum physics, this is called Entanglement Entropy.
• The Result: When you wiggle the edge, the entanglement changes in a very specific way.

2. The "Gravity" World (The AdS2/Bath System)

Now, imagine a different setup. You have a flat floor (the "Bath") and a small, curved, trampoline-like patch of space (the "Island" or "AdS2 region") glued to it.

• The Setup: The flat floor has no gravity. The trampoline patch has gravity (specifically, a type called Jackiw-Teitelboim or JT gravity).
• The Action: You don't wiggle the edge. Instead, you change the "tension" or shape of the trampoline patch (this is controlled by a field called the dilaton).
• The Measurement: You measure the entanglement between the flat floor and the trampoline. In this gravity world, the answer isn't just a simple sum; it involves finding a secret "Island" of information hidden inside the trampoline. This is called the Island Formula.

The "Aha!" Moment: They Are Twins

The authors did the math for both worlds.

• In the Flat World, they calculated how the entanglement changed when they wiggled the edge.
• In the Gravity World, they calculated how the entanglement changed when they tweaked the gravity patch.

The Result: The numbers matched perfectly.

The Analogy:
Think of it like two different languages describing the same song.

• Language A (BCFT): "I am pushing the edge of the fabric."
• Language B (Gravity): "I am stretching the trampoline."

The paper proves that these two sentences are actually saying the exact same thing. If you know how to wiggle the edge, you automatically know how to stretch the trampoline, and vice versa.

Why Is This a Big Deal?

Usually, when physicists say "gravity emerges from quantum mechanics," they are talking about Holography (like a 3D hologram popping out of a 2D surface). This usually requires the system to be very special and "heavy" (holographic).

The Twist in This Paper:
The authors found that you don't need a special, heavy holographic system for this to work.

• They showed that even for "lighter," more ordinary quantum systems (as long as they have a lot of "stuff" in them, known as a large central charge), the edge wiggle still creates a gravity-like effect.
• They introduced a concept called "Weak Vacuum Block Dominance."
  • Analogy: Imagine a choir. Usually, to get a perfect harmony (gravity), you need a very specific, rare choir (Holographic). The authors found that even a choir with many different singers (non-holographic), as long as the "lead singer" (the vacuum) is loud enough, the harmony still sounds like gravity.

The "Island" Concept

A major part of modern physics (solving the Black Hole Information Paradox) involves the idea of an Island.

• Imagine you are trying to read a book (the quantum system).
• Usually, you only read the pages in front of you.
• But if gravity is involved, the book might have a secret "Island" of pages hidden inside the cover that you didn't know existed.
• To get the right answer, you have to count the pages you see plus the secret island pages.

The paper shows that when you wiggle the edge of the quantum fabric, it's mathematically equivalent to suddenly discovering that secret island of pages in the gravity version of the story.

Summary in One Sentence

This paper proves that wiggling the edge of a quantum system is mathematically identical to creating a tiny patch of gravity underneath it, and this trick works even for systems that aren't the "super-special" kind usually required for gravity to appear.

Why Should You Care?

This is a step toward understanding how gravity emerges from quantum mechanics. It suggests that gravity might not be a fundamental force of nature, but rather a side effect of how quantum information is organized at the edges of our universe. It's like realizing that the "bumps" on a rug aren't real bumps, but just the result of how the threads are woven together.

Technical Summary

1. Problem Statement

The paper investigates the relationship between boundary deformations in two-dimensional Boundary Conformal Field Theories (BCFTs) and gravitational descriptions. Specifically, it addresses whether the von Neumann entanglement entropy of subregions in a BCFT with an infinitesimally deformed boundary can be reproduced by a gravitational computation involving Jackiw-Teitelboim (JT) gravity.

Previous work (e.g., [26, 27]) established that in holographic BCFTs (dual to AdS$_3$ with an End-of-the-World brane), boundary displacements correspond to non-trivial dilaton profiles in an effective 2D JT gravity theory living on the brane. However, these results relied on the assumption that the BCFT is holographic (dual to a higher-dimensional bulk with a sparse spectrum).

The central question: Can this correspondence hold for non-holographic large-$c$ BCFTs? Does the emergence of JT gravity as an effective description of boundary deformations require the strict constraints of holography, or can it arise under weaker conditions?

2. Methodology

The authors employ a "bottom-up" approach, comparing entanglement entropies calculated in two distinct systems without assuming a higher-dimensional bulk dual:

1. The BCFT System:

   • A 2D BCFT on the upper half-plane ($H^+$) with a boundary at $\text{Im}(z)=0$.
   • The boundary is deformed by an infinitesimal global conformal transformation (a broken symmetry of the BCFT).
   • Entanglement entropy is computed for single and multiple intervals using the replica trick and conformal block expansions.
   • The deformation is treated as a linear perturbation sourced by the displacement operator $D(\tau) = T_{nn}(\tau)$.

2. The AdS$_2$/Bath System:

   • A 2D CFT living on the complex plane, split into a non-gravitating "bath" (upper half-plane, $H^+$) and a gravitating region (lower half-plane, $H^-$).
   • The gravitating region is governed by JT gravity coupled to the CFT.
   • The boundary between the bath and the gravity region is transparent.
   • Entanglement entropy is computed using the Quantum Extremal Island (QEI) formula.
   • Crucially, the dilaton profile $\phi$ in the JT region is chosen such that its asymptotic behavior matches the magnitude of the BCFT boundary deformation.

Key Comparison: The authors calculate the von Neumann entropy in the deformed BCFT and compare it to the island entropy in the AdS$_2$/bath system. They verify if the two match under the identification of the boundary deformation parameter with the asymptotic dilaton value.

3. Key Contributions and Results

A. Single Interval Matching (Zero and Finite Temperature)

• Result: The authors explicitly demonstrate that the linear correction to the entanglement entropy in a BCFT with a deformed boundary exactly matches the island entropy in the AdS$_2$/bath system.
• Mechanism:
  • In the BCFT, the deformation shifts the entropy by a term proportional to the displacement operator expectation value.
  • In the gravity dual, this shift corresponds to the value of the dilaton field $\phi$ at the quantum extremal surface (the island endpoint).
  • The matching condition is $\Delta X(\tau) = \lim_{y\to 0} y \phi(\tau, y)$, linking the boundary displacement $\Delta X$ to the dilaton.
• Boundary Entropy: The constant term in the dilaton (related to the topological term in JT gravity, $\phi_0$) is identified with the boundary entropy $\log g_b$ of the BCFT, including renormalization effects from matter fields.

B. Multi-Interval Analysis and Spectral Conditions

For single intervals, conformal symmetry alone fixes the result. To test the robustness of the correspondence, the authors analyze two intervals and multiple intervals, where the entropy depends on the full operator spectrum and OPE coefficients.

• Bulk vs. Boundary Channels:
  • Bulk Channel (Bulk OPE): Corresponds to the limit where intervals are far from the boundary. The result matches the gravity calculation without islands. This requires the vacuum block dominance in the bulk channel.
  • Boundary Channel (Boundary OPE): Corresponds to the limit where intervals are close to the boundary. The result matches the gravity calculation with islands.
• Weak Vacuum Block Dominance:
  • Standard holographic CFTs require a sparse spectrum (few light operators) to ensure vacuum block dominance.
  • The authors show that for their correspondence to hold, the BCFT does not need to be holographic. Instead, it satisfies a condition they term "Weak Vacuum Block Dominance":
    1. Heavy operators ($\Delta \sim O(c)$) must be exponentially suppressed in the relevant kinematic limits.
    2. The sum of OPE coefficients for light operators must not grow exponentially with $c$ (specifically, $\sum B^2 \sim O(1)$ or subleading relative to the vacuum contribution in the large-$c$ limit).
  • This allows for theories with an exponentially large number of light operators (non-holographic), provided their OPE coefficients are sufficiently small.

C. Generalization to Multiple Intervals

The authors extend the analysis to $s$ intervals. They show that the entanglement entropy in a specific fusion channel of the BCFT matches the island entropy in the corresponding channel of the AdS$_2$/bath system, provided the "Weak Vacuum Block Dominance" conditions are met. The correction due to boundary deformation appears as a sum of single-point dilaton contributions evaluated at the island endpoints.

4. Significance and Implications

1. Universality of JT Gravity: The paper provides strong evidence that JT gravity emerges as the universal effective description of boundary deformations in large-$c$ BCFTs, independent of whether the theory is holographic. This suggests that the connection between boundary dynamics and 2D gravity is more fundamental than previously thought.
2. Relaxation of Holographic Constraints: By introducing "Weak Vacuum Block Dominance," the authors demonstrate that the emergence of a gravitational dual (specifically JT gravity) does not strictly require the sparse spectrum typical of holographic CFTs. This opens the door to studying gravitational phenomena in non-holographic quantum systems.
3. Island Formula Validation: The work reinforces the validity of the Quantum Extremal Island formula as a tool for computing entanglement in open systems coupled to gravity, even when the "bath" is a standard CFT and the "gravity" region is a lower-dimensional effective theory.
4. Connection to Double Holography: The results offer a purely CFT-based derivation of the findings from double-holographic setups (where an ETW brane in AdS$_3$ is dual to JT gravity), confirming that the physics of the brane displacement can be captured entirely within the 2D boundary theory.

Conclusion

Neuenfeld and Tellinger successfully demonstrate that the entanglement entropy of deformed BCFTs at large central charge is reproduced by JT gravity coupled to a bath. They establish that this correspondence relies on "Weak Vacuum Block Dominance," a condition significantly weaker than the strict holographic requirements. This suggests that the emergence of 2D gravity from boundary deformations is a robust feature of large-$c$ CFTs, potentially applicable to a broader class of quantum systems than just those with higher-dimensional holographic duals.