Two Splits, Three Ways: Advances in Double Splitting Quenches

This paper introduces a new holographic method for calculating entanglement entropy in 1+1d CFTs with multiple boundaries, successfully reproducing known results for double-split quenches while establishing a framework for future studies of systems with more than two splits and nonzero temperatures.

The Gist

The Big Picture: Shattering a Quantum Glass

Imagine you have a perfectly smooth, invisible sheet of rubber representing the entire universe of a specific quantum system. This sheet is in its most calm, relaxed state (the "ground state").

Now, imagine you take a pair of scissors and instantly snip this sheet in two places at the exact same time. Suddenly, your one big sheet is cut into three separate, floating pieces. These pieces stop talking to each other, but they are still vibrating and interacting internally.

This is what physicists call a "Splitting Quench." It's a theoretical model for what happens when a super-hot, messy soup of particles (like the quark-gluon plasma created in heavy ion collisions) suddenly cools down and shatters into individual particles (hadrons) that fly apart.

The big question the authors ask is: How does the "connection" (entanglement) between these pieces change over time? In quantum mechanics, even if you cut things apart, they remain strangely linked. The authors want to calculate exactly how strong that link is and how it evolves.

The Problem: A Messy Map

To solve this, the authors use a powerful tool called Holography. Think of this like a 3D movie projector.

• The Screen (2D): The quantum world where the cuts happen.
• The Projector (3D): A higher-dimensional "bulk" space (like a black hole geometry) that projects the physics of the screen.

The problem is that when you cut the sheet twice, the shape of the "screen" becomes very weird and twisted. It's like trying to draw a map of a crumpled piece of paper onto a flat table. The math gets incredibly messy, especially when you want to add more cuts later.

The Solution: Three Ways to Flatten the Paper

The authors' main achievement is showing that you can solve this messy math problem using three different methods (or "maps"), and they all lead to the exact same answer. This proves their new method is solid.

Here are the three ways they "flattened" the crumpled paper:

1. The "Theta Function" Map (The Old Way)

• The Analogy: Imagine you have a complex, twisted knot. The first method uses a very specific, pre-made instruction manual (a "Theta Function") to untie it.
• The Catch: This manual is huge, complicated, and hard to read. If you wanted to add a third or fourth cut to the paper, this manual would become impossibly long and confusing. It works for two cuts, but it's not scalable.

2. The "Abel-Jacobi" Map (The Shortcut)

• The Analogy: Instead of untangling the knot piece by piece, this method realizes the knot is actually just a donut (a torus) that got squished.
• How it works: It takes the twisted sheet and wraps it around a donut shape. Because a donut is a very regular, predictable shape, the math becomes much simpler. It's like realizing that a complicated maze is actually just a circle.
• Why it's cool: This is the "inverse" of the first method. It's much easier to calculate and, most importantly, it's easy to add more cuts later. You just make the donut more complex (add more holes), and the math stays manageable.

3. The "Schottky Uniformization" Map (The New Tool)

• The Analogy: Imagine you have a rubber sheet with holes in it. This method says, "Let's cut out a perfect circle from a different sheet of rubber and stretch our messy sheet over it."
• How it works: They use a special mathematical function (the Schottky-Klein prime function) to map the messy, cut-up world onto a clean, circular domain (an annulus).
• Why it's cool: This is the authors' "novel" contribution. It's a fresh way of looking at the problem that connects directly to the 3D holographic world. It allows them to build a bridge from the messy 2D cuts to the clean 3D geometry of a black hole.

The Result: A Perfect Match

The authors ran the numbers for all three methods.

• They calculated how the "entanglement entropy" (a measure of how connected the pieces are) grows over time.
• They looked at four different types of "subsystems" (different ways of grouping the cut pieces).
• The Verdict: All three methods produced identical results.

This is a huge deal. It means their new "Schottky" method is just as accurate as the old, trusted method, but it's built on a foundation that can easily handle many cuts, not just two.

Why Should You Care?

1. Future Proofing: The authors are essentially building a better engine. The old engine (Theta functions) works for small cars (two cuts). Their new engine (Schottky/Abel-Jacobi) is built to handle trucks with many trailers (many cuts).
2. Real World Physics: While this is theoretical math, it helps us understand how the universe "breaks apart." When a heavy ion collision happens in a particle accelerator, the debris flies apart. This model helps physicists understand the quantum "glue" that holds the debris together right before it flies off.
3. The "Multifragmentation" Dream: The authors hope to use this to create a simplified model of how a chaotic quantum system (like the early universe or a particle collision) fragments into many distinct pieces.

In a Nutshell

The paper is about taking a very difficult math problem (calculating quantum connections after a double cut) and showing that there are three ways to solve it. They proved two new ways work just as well as the old way, but the new ways are much easier to use and can be expanded to solve even harder problems in the future. They successfully mapped a messy quantum cut-up to a clean 3D holographic picture, confirming their new tools are ready for the next big challenge.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the theoretical challenge of modeling the multifragmentation of strongly interacting quantum systems. Specifically, it investigates the dynamics of entanglement entropy in a $(1+1)$-dimensional Conformal Field Theory (CFT) when its ground state is instantaneously split into multiple non-interacting segments.

• Physical Context: This models scenarios like relativistic heavy-ion collisions where a quark-gluon plasma (initially a pure, low-entropy state) fragments into hadrons that cease to interact with each other but remain internally interacting.
• Theoretical Gap: While "joining quenches" (merging two half-lines) and "single split quenches" are well-understood, calculating the entanglement entropy for multiple splits (specifically $n > 1$ cuts) on an infinite line is mathematically complex.
• Specific Goal: The authors aim to develop a robust method to calculate holographic entanglement entropy (HEE) for a CFT split into three segments via two cuts (a "double split"). They seek to reproduce existing results for this case and establish a framework generalizable to $n$ cuts and non-zero temperatures.

2. Methodology

The core of the work involves calculating the HEE using the Ryu-Takayanagi (RT) prescription, which relates the entanglement entropy of a boundary subsystem to the area (length, in 2D) of the minimal geodesic in the bulk AdS geometry.

Since the boundary geometry is a complex plane with cuts (a non-simply connected Riemann surface), the authors employ three distinct conformal mapping techniques to map this complicated worldsheet to a simpler domain where the bulk geometry is known (specifically, the BTZ black hole geometry).

A. The Three Conformal Maps

The authors demonstrate that three different mathematical approaches yield identical physical results:

1. Modified Theta Function Map (Review):

   • Based on previous work by Caputa et al. [2].
   • Maps the doubly cut plane to a rectangle with identified sides (an annulus/torus).
   • Uses a truncated modified theta function.
   • Limitation: Generalizing to $n > 2$ cuts requires generalized theta functions, which are computationally cumbersome.

2. Abel-Jacobi Map (Novel Application):

   • The authors interpret the two-cut plane as the projection of an elliptic curve (genus 1 Riemann surface).
   • They utilize the Schottky double (a branched double cover) to treat the surface as a compact torus.
   • The map is the inverse of the theta function map: it maps the elliptic curve to its Jacobian variety (a torus defined by periods of canonical differentials).
   • Advantage: This approach naturally generalizes to hyperelliptic curves (more cuts) and simplifies the calculation of derivatives required for the holographic metric.

3. Schottky Uniformization (Novel Prescription):

   • This is the primary novel contribution. The authors map the worldsheet to a Schottky domain (a unit disk with $g$ circles removed, where $g$ is the genus).
   • They utilize the Schottky-Klein prime function to construct the explicit conformal map from the annulus (Schottky domain) to the plane with cuts.
   • The bulk geometry is constructed by quotienting Euclidean $AdS_3$ by the Schottky group (a subgroup of $SL(2, \mathbb{C})$).
   • Advantage: Provides a systematic way to handle arbitrary numbers of cuts and boundaries, laying the groundwork for future multi-fragmentation studies.

B. Holographic Calculation

For all three maps, the procedure is:

1. Identify the conformal map $w(\eta)$ from the boundary worldsheet coordinates ($w$) to the uniformized coordinates ($\eta$).
2. Use the Schwarzian derivative of this map to determine the stress-energy tensor and the bulk metric (specifically the BTZ metric).
3. Calculate the geodesic length in the bulk connecting the endpoints of the subsystem.
4. Apply the RT formula: $S = \frac{c}{6} \ln(\frac{L_{\gamma}}{\epsilon})$, considering both connected and disconnected geodesic phases to find the minimal entropy.

3. Key Contributions

• Validation of Schottky Uniformization: The paper successfully applies the Schottky-Klein prime function formalism to a holographic entanglement entropy problem, proving it yields the same results as established methods.
• Equivalence of Methods: The authors provide a rigorous cross-check by showing that the Theta function, Abel-Jacobi, and Schottky uniformization approaches produce perfect numerical agreement for the entanglement entropy growth.
• Generalization Framework: By utilizing the Abel-Jacobi map and Schottky uniformization, the authors demonstrate a pathway to calculate entanglement entropy for systems with more than two cuts ($n > 2$), which was previously intractable using standard theta function truncations.
• Explicit Geodesic Derivations: The paper includes detailed derivations of geodesic lengths in the BTZ geometry under various coordinate systems (Appendix B), clarifying the relationship between UV cutoffs and bulk regulators.

4. Results

The authors performed numerical calculations for four qualitatively different subsystems ($A_1, A_2, A_3, A_4$) defined by their positions relative to the two cuts at $x = \pm b$:

• $A_1$: To the right of both cuts.
• $A_2$: Between the two cuts.
• $A_3$: Split by one cut.
• $A_4$: Includes both cuts.

Findings:

• The time evolution of entanglement entropy ($\Delta S_A$) was calculated for all three methods.
• The results for connected (direct geodesic) and disconnected (geodesic touching the boundary/horizon) phases matched perfectly across all three approaches.
• The transition time between connected and disconnected phases was consistent with earlier literature [2].
• The results confirm that the holographic dual of a double-split CFT is effectively a BTZ black hole geometry, with the specific metric determined by the conformal map's Schwarzian derivative.

5. Significance and Future Outlook

• Theoretical Robustness: The work solidifies the connection between Riemann surface theory (specifically Schottky groups and uniformization) and AdS/CFT correspondence. It proves that complex multi-boundary problems can be solved using standard uniformization techniques.
• Heavy Ion Collision Modeling: The authors propose that this formalism can be used to build schematic models of multifragmentation in heavy-ion collisions. By analyzing the entanglement structure among multiple "hadron" segments, one can better understand the thermalization and entropy production in the quark-gluon plasma.
• Future Directions:
  • Explicit calculation of entanglement entropy for $n > 2$ splits.
  • Extension to non-zero temperature systems.
  • Application to local projection measurements and other non-simply connected worldsheet scenarios.

In summary, "Two Splits, Three Ways" serves as a foundational proof-of-concept, demonstrating that Schottky uniformization and Abel-Jacobi maps are powerful, generalizable tools for calculating holographic entanglement entropy in complex, multi-segmented quantum systems.