Jones index from Rényi entropies in the Ising conformal field theory

This paper investigates the relationship between the Jones index and Rényi entropies in the Ising and free Majorana fermion conformal field theories, deriving analytic expressions for crossing asymmetry and demonstrating that the Jones global index emerges from the leading term of this asymmetry in the limit of adjacent intervals.

The Gist

Imagine you are standing in a vast, infinite library. This library represents the quantum world. Inside, there are books (particles and fields) and rules for how they can be combined.

In a "perfect" library (what physicists call a Complete Model), every single book you can possibly write using the available alphabet is already on the shelves. If you try to build a new story, you'll find it's just a rearrangement of existing books.

However, sometimes we look at a sub-library (a Submodel). This is a smaller section of the big library. In this smaller section, some books are missing. You might have the alphabet, but you can't write certain stories because the specific "magic words" (charged particles) needed to build them are locked away in the main library.

This paper is about measuring how much information is missing from these sub-libraries. The authors use a very specific mathematical tool called the Jones Index to do this. Think of the Jones Index as a "Missing Book Score."

• A score of 1 means the sub-library is actually the whole library (nothing is missing).
• A score of 4, 16, or higher means a lot of books are missing, and the library is "incomplete."

The Detective's Tool: The "Crossing Asymmetry"

How do you measure missing books without walking through every aisle? The authors use a clever trick involving entanglement.

Imagine you take two separate, empty rooms in the library (two disjoint intervals). You ask: "How much do these two rooms 'know' about each other?" In quantum physics, this is measured by something called Rényi Entropy.

Usually, if the library is complete, the relationship between these two rooms is perfectly symmetrical. It doesn't matter which way you look at them; the information flow is balanced.

But if the library is incomplete (missing those magic words), the symmetry breaks. The relationship becomes "lopsided." The authors call this lopsidedness the "Crossing Asymmetry."

• The Analogy: Imagine two people talking through a wall. If the wall is solid and perfect, they hear each other equally well. If the wall has a hidden, one-way door (the missing information), the conversation becomes weird. One person hears more than the other, or the timing is off.
• The Measurement: The authors found that if you measure this "weirdness" (the asymmetry) when the two rooms are right next to each other, the size of the weirdness tells you exactly the Jones Index (the Missing Book Score).

The Two Case Studies

The paper tests this idea on two specific types of quantum libraries:

1. The Ising Model (The Spin Library):

   • This is a famous model representing tiny magnets (spins) that can point up or down.
   • The authors looked at different "sections" of this library.
   • Section A: Contains only "Up" and "Down" spins. It's missing the "Spin Flip" book. The Missing Book Score (Jones Index) is 4.
   • Section B: Contains only the "No Spin" book. It's missing almost everything. The Score is 16.
   • The Result: By measuring the "Crossing Asymmetry" of the empty rooms in these sections, the math perfectly predicted the scores of 4 and 16.

2. The Free Majorana Fermion (The Ghost Library):

   • This is a library of "ghost" particles that are their own antiparticles.
   • It has a different set of missing books (specifically, it's missing books that allow you to swap left and right).
   • The authors found that even though this library looks different, the "Crossing Asymmetry" trick still works, revealing its own unique Missing Book Scores.

The "Magic" of the Limit

The most beautiful part of the paper is a mathematical limit. The authors show that you can calculate this "Missing Book Score" using a variable called $n$ (the Rényi index), which is like a "zoom level" or a "complexity setting."

• You can calculate the asymmetry for $n=2$, $n=3$, $n=10$, or even $n=100$.
• The paper proves that no matter what number you pick for $n$, if you look at the "Crossing Asymmetry" when the two rooms are adjacent, the leading term of the math always points to the same answer: the Jones Index.

It's like having a thousand different types of rulers (different $n$ values), and no matter which one you use, they all measure the length of the missing books to be exactly the same number.

Why Does This Matter?

In the past, calculating these "Missing Book Scores" required very difficult, abstract algebra. This paper shows a new, more physical way to find them using entanglement (how quantum parts of a system are connected).

It's like discovering that instead of counting every missing book in a library by hand, you can just stand in the hallway, listen to the echo between two rooms, and instantly know exactly how many books are missing from the entire building.

In short: The authors found a universal "echo test" (Crossing Asymmetry) that reveals the hidden structure and completeness of quantum theories, proving that the way quantum information is shared between two points tells us exactly how "broken" or "incomplete" the underlying rules of that universe are.

Technical Summary

1. Problem Statement

The paper investigates the relationship between algebraic Quantum Field Theory (QFT) concepts and Quantum Information Theory measures, specifically focusing on the Jones index and Rényi entropies.

• Context: In Algebraic QFT, the Haag duality condition ($A(R) = A'(R')$) often holds for complete theories. However, for submodels (theories with restricted spectra or non-trivial superselection sectors), Haag duality is violated. This violation is quantified by the Jones index ($\mu$), which measures the "size" of the inclusion of the additive algebra into the maximal algebra.
• The Gap: While it was previously established that the global Jones index can be extracted from the crossing asymmetry of Rényi entropies in the limit where two disjoint intervals become adjacent ($\xi \to 1$), this relation was only explicitly verified for the case where the Rényi index $n=2$.
• Objective: The authors aim to generalize this relation for a generic finite Rényi index $n \geq 2$. They specifically test this on the Ising CFT$_2$ (a rational CFT with $c=1/2$) and the free Majorana fermion, analyzing their submodels that violate Haag duality.

2. Methodology

The authors employ a combination of Algebraic QFT, Conformal Field Theory (CFT), and Riemann surface techniques.

• Crossing Asymmetry ($A_{T,n}$): They utilize the quantity defined as:
  $$A_{T,n}(\xi) \equiv \frac{1}{n-1} \log \left( \frac{F_{T,n}(\xi)}{F_{T,n}(1-\xi)} \right)$$
  where $F_{T,n}(\xi)$ is the model-dependent function in the Rényi entropy formula for two disjoint intervals, and $\xi$ is the cross-ratio.
• Riemann Surface Partition Functions: The Rényi entropy for $n$ copies is mapped to the partition function of the CFT on a Riemann surface of genus $g = n-1$. The authors express these partition functions using Riemann theta functions with characteristics ($\Theta[\mathbf{p}, \mathbf{q}](\tau)$).
• Submodel Construction via Projectors: To study non-complete submodels (e.g., those generated only by the identity and energy operator, or only the identity), the authors construct partition functions by inserting projectors (related to Verlinde lines or topological defect lines) into the trace of the complete theory.
  • For the Ising model, they isolate submodels $T_{Z_2}$ (generated by $\{1, \epsilon\}$) and $T_{su(2)_2}$ (generated by $\{1\}$).
  • For the free Majorana fermion, they analyze submodels with fermionic parity constraints ($T_{Z_2^L}, T_{Z_2^R}$).
• Analytic and Numerical Analysis:
  • They derive explicit analytic expressions for the crossing asymmetry for arbitrary $n$.
  • They perform asymptotic expansions as $\xi \to 0$ (large separation) and $\xi \to 1$ (adjacent intervals).
  • They perform numerical evaluations for $n$ up to 8 to verify the analytic limits and study monotonicity.

3. Key Contributions

1. Generalization to Arbitrary $n$: The paper provides the first explicit analytic expressions for the crossing asymmetry $A_{T,n}(\xi)$ for generic integer $n \geq 2$ in the Ising CFT$_2$ and free Majorana fermion models.
2. Verification of the Jones Index Relation: They rigorously demonstrate that the limit of the crossing asymmetry as the intervals become adjacent ($\xi \to 1$) yields the global Jones index $\mu_T$ for any finite $n$:
   $$\lim_{\xi \to 1} A_{T,n}(\xi) = -\lim_{\xi \to 0} A_{T,n}(\xi) = \frac{1}{2} \log \mu_T$$
   This confirms the conjecture from [26] for generic $n$.
3. Modular Group Analysis: The authors analyze the invariance properties of the partition functions under the symplectic modular group $Sp(2g, \mathbb{Z})$. They identify that the submodels are invariant under a specific Siegel parabolic subgroup ($GL(g, \mathbb{Z}) \ltimes Sym(g, \mathbb{Z})$), which preserves the condition $p=0$ in the theta function characteristics. The violation of Haag duality corresponds to the lack of invariance under the transformation $\xi \to 1-\xi$ (which maps to a specific modular transformation outside this subgroup).
4. Non-Commuting Limits: They investigate the limits $n \to \infty$ (single-copy entanglement) and $\xi \to 0$. They find that these limits do not commute, as the subleading terms in the expansion of the crossing asymmetry depend on $n$ in a way that diverges as $n \to \infty$.

4. Key Results

• Ising CFT$_2$ Submodels:
  • Complete Model ($T_{Ising}$): $\mu = 1$ (Haag duality holds).
  • Submodel $T_{Z_2}$ (generated by $\{1, \epsilon\}$): The crossing asymmetry leads to $\mu_{Z_2} = 4$.
  • Submodel $T_{su(2)_2}$ (generated by $\{1\}$): The crossing asymmetry leads to $\mu_{su(2)_2} = 16$.
  • Non-Closed Subalgebra: They show that a subalgebra generated by $\{1, \sigma\}$ (which is not closed under fusion) yields a "Jones index" of $16/9$, which is not an allowed value (must be $\geq 1$ and satisfy specific discrete sequences or be $\geq 4$), confirming it is not a valid local subalgebra.
• Free Majorana Fermion:
  • The complete model has $\mu=1$.
  • Submodels $T_{Z_2^L}$ and $T_{Z_2^R}$ (involving chiral fermions) yield $\mu = 4$.
  • The submodel $T_{su(2)_2}$ yields $\mu = 16$.
• Asymptotic Behavior:
  • The leading term of the expansion of $A_{T,n}(\xi)$ near $\xi \to 1$ is constant and equal to $\frac{1}{2}\log \mu_T$.
  • The subleading term depends on the Rényi index $n$ and the conformal dimensions of the lightest non-local fields in the spectrum.
• Monotonicity: Numerical results for $2 \leq n \leq 5$ suggest that the crossing asymmetry is a monotonically decreasing function of $\xi$ (vanishing at $\xi=1/2$), though a rigorous proof for generic $n$ remains an open problem.

5. Significance

• Bridge Between Fields: The work solidifies the connection between the algebraic structure of QFT (Jones index, superselection sectors) and entanglement measures (Rényi entropies), showing that entanglement data can fully reconstruct algebraic invariants.
• Diagnostic Tool: It provides a practical method to detect and quantify Haag duality violations (and thus the presence of non-trivial superselection sectors) in CFTs using only entanglement entropy data, without needing to explicitly construct the algebraic endomorphisms.
• Validation of Modular Constraints: The analysis highlights how modular invariance on higher-genus Riemann surfaces is directly linked to the completeness of the theory. The failure of specific modular transformations for submodels is the mathematical signature of their incompleteness.
• Future Directions: The paper opens avenues for studying these relations in thermal states, excited states, and other CFTs (like compact bosons), and exploring connections to Renormalization Group flows.

In summary, the paper successfully extends the theoretical framework linking Jones indices to Rényi entropies to arbitrary $n$, providing explicit analytic formulas and numerical verification for the Ising and free fermion models, thereby deepening the understanding of entanglement in algebraic QFT.