Bootstrapping non-unitary CFTs

This paper introduces a unitarity-independent evolutionary algorithm that bootstraps the spectra of general conformal field theories by iteratively optimizing trial spectra to maximize the satisfaction of crossing symmetry, successfully reproducing known minimal models for Virasoro blocks with $c<1$.

The Gist

Imagine you are trying to solve a massive, multi-dimensional jigsaw puzzle. But there's a catch: you don't have the picture on the box, and you aren't even sure if the pieces you have are the "right" kind of pieces. In the world of theoretical physics, this puzzle is called a Conformal Field Theory (CFT). It's a mathematical framework used to describe how particles and forces behave at the most fundamental levels, especially when things are perfectly balanced (like in a critical phase transition).

For decades, physicists have been trying to solve this puzzle using a method called the Bootstrap. The name comes from the idea of "pulling yourself up by your own bootstraps"—solving the system using only its own internal rules, without needing external data.

The Old Way: The "Good Boy" Filter

Traditionally, the Bootstrap method worked like a strict teacher with a very specific rule: "Everything must be positive."

In physics, this rule is called Unitarity. It basically means that probabilities must add up to 100% and nothing can have "negative probability" (which is physically impossible in our everyday world). Because of this rule, the math became very neat and tidy, like a convex bowl. Physicists could use powerful computers to find the "bottom" of the bowl, which represented the correct solution.

However, this method had a big blind spot: It couldn't see the "bad" solutions.
There are many interesting theories in physics (like those describing certain chaotic systems or "complex" fixed points) that are non-unitary. In these theories, the math allows for "negative probabilities" or complex numbers. The old "strict teacher" method would immediately throw these theories out, even though they might be the correct description of reality in those specific contexts. It was like trying to find a solution to a puzzle, but only looking at pieces that were blue, ignoring all the red and green ones that might actually fit.

The New Idea: The "Stability Test"

In this new paper, the authors propose a clever new strategy to find those "non-unitary" solutions. They stop trying to force the pieces to be positive and instead ask a different question: "Is this solution stable?"

Here is the analogy:

Imagine you are trying to tune a radio to find a specific station.

• The Old Way: You only listen to stations that play "happy music" (Unitary). If a station plays "sad music" (Non-unitary), you ignore it.
• The New Way: You turn the dial to a specific frequency. If you are exactly on the station, the music sounds perfect and clear, no matter how you slightly wiggle the dial. But if you are off the station, the music gets fuzzy, crackly, and changes wildly with the slightest movement.

The authors realized that in the math of the Bootstrap, the "fuzziness" is the key.

1. The Setup: They pick a guess for the "spectrum" (the list of particles/weights in the theory).
2. The Test: They calculate the "OPE coefficients" (which are like the volume knobs for how much each particle contributes) using different mathematical "viewpoints" (called cross-ratios).
3. The Result:
   • If the guess is wrong, the volume knobs will fluctuate wildly depending on which viewpoint you use. The solution is "unstable."
   • If the guess is right (or very close to right), the volume knobs stay perfectly steady, no matter how you change the viewpoint. The solution is "statistically stable."

How They Did It

The authors built a computer program (using a smart algorithm called CMA-ES, which is like a digital evolution that learns by trial and error) to search for these stable solutions.

• Step 1: They threw a dart at the board (random guess).
• Step 2: They checked if the "volume knobs" were stable.
• Step 3: If the knobs wobbled, the computer evolved the guess to make them steadier.
• Step 4: They repeated this until they found a set of numbers where the knobs didn't wobble at all.

What They Found

1. Recovering the Classics: First, they tested their method on theories they already knew the answers to (the "A-series minimal models"). Even though their method didn't assume the "positive probability" rule, it still found the correct answers. This proved the method works.
2. Finding the Unknown: Then, they looked for theories with a central charge ($c$) greater than 1, which are usually very hard to solve. They found stable, truncated solutions that look like real physical theories, even though they are non-unitary.

Why This Matters

This is a big deal because it opens the door to a whole new world of physics.

• Before: We could only map the "safe" part of the universe (where probabilities are positive).
• Now: We have a map-making tool that can explore the "dangerous" or "weird" parts of the universe (where complex numbers and negative probabilities exist).

It's like having a new pair of glasses that lets you see colors that were previously invisible. The authors have shown that even without the strict rules of "good behavior" (unitarity), the universe still has a hidden order (stability) that we can find if we know how to look for it.

In a nutshell: They stopped asking "Is this solution positive?" and started asking "Is this solution consistent?" This simple shift allowed them to find new, weird, and fascinating solutions to the universe's biggest puzzles.

Technical Summary

1. Problem Statement

The numerical conformal bootstrap has traditionally relied on unitarity (positivity of OPE coefficients) to formulate the search for Conformal Field Theory (CFT) data as a convex optimization problem. This approach successfully constrains unitary theories (e.g., 3D Ising model) but faces two major limitations:

1. Non-unitary theories: Many physically interesting systems, such as renormalization-group flows near complex fixed points and certain holographic settings, are non-unitary. The standard convex formulation breaks down because OPE coefficients can be negative or complex, removing the convexity required for rigorous bounds.
2. Interior solutions: Optimization-based methods are excellent at finding the boundaries of the allowed theory space but struggle to locate specific theories residing in the interior of the parameter space.

Existing non-convex search strategies (Monte Carlo, machine learning) often require significant prior knowledge or struggle with the high dimensionality of the search space (scaling dimensions $\Delta$, spins $s$, and OPE coefficients $f$).

2. Methodology

The authors propose a new strategy that bypasses the need for unitarity by treating the bootstrap as a search for statistical stability in inferred OPE data.

Core Concept: Inversion and Stability

Instead of scanning over the full space of $(\Delta, s, f^2)$, the method fixes a trial spectrum (a set of scaling dimensions $\Delta_k$ and spins $s_k$) and inverts the crossing equations to solve for the OPE coefficients ($C_k = f_{\phi\phi k}^2$).

• Exact Solution: For a true CFT, the inferred OPE coefficients are independent of the choice of cross-ratios $(z, \bar{z})$.
• Approximate/Truncated Solution: If the spectrum is truncated or incorrect, the inferred coefficients fluctuate as the evaluation points $(z, \bar{z})$ vary.

The Objective Function

The authors define a scalar objective function $R$ based on the statistical spread of the inferred coefficients across a set of sampled cross-ratios $\{z_j\}$.

1. Matrix Formulation: The crossing equations are discretized into a linear system $G \cdot C + e = v$, where $G$ contains conformal blocks, $C$ are the squared OPE coefficients, and $e$ is the truncation error.
2. Least Squares Inference: By sampling $N_z \gg N$ points (where $N$ is the number of operators in the truncation), the system is over-determined. The coefficients $C$ are inferred via least squares.
3. Reward Metric: The objective function maximizes the stability of these coefficients:
   $$R = -\sum_{i=1}^{N_r} \log \left| \frac{\sigma(C_i)}{\text{Mean}(C_i)} \right|$$
   where $\sigma$ is the standard deviation of the inferred coefficient $C_i$ across the sampled points. A high $R$ indicates that the inferred data is stable (low variance), implying the spectrum satisfies crossing symmetry well.

Optimization Algorithm

• Search Space: The optimization is performed only over the spectrum ($\Delta_k, s_k$), removing the OPE coefficients from the nonlinear search space.
• Optimizer: The authors use the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), a derivative-free stochastic optimizer suitable for non-convex, noisy landscapes.
• Implementation: The calculation of Virasoro conformal blocks and the reward function is highly parallelized on GPUs to handle the computational cost of evaluating thousands of points per iteration.

3. Key Contributions

• Non-Unitary Bootstrap Framework: Introduces a practical, non-convex method to search for CFT solutions without assuming positivity of OPE coefficients.
• Statistical Stability Criterion: Demonstrates that the variance of inferred OPE coefficients across different cross-ratios serves as a robust proxy for the "distance" to a true solution.
• Dimensionality Reduction: Effectively reduces the search space by analytically solving for OPE coefficients given a spectrum, turning a complex non-linear problem into a search over scaling dimensions alone.
• GPU-Accelerated Search: Implements a fully GPU-parallelized pipeline using CMA-ES, enabling efficient exploration of high-dimensional parameter spaces.

4. Results

The method was tested on 2D CFTs using Virasoro conformal blocks.

A. Reproduction of Known Minimal Models ($c < 1$)

• Exact Spectra: In regions where the exact spectrum fits within the truncation (e.g., A-series minimal models with 1, 2, or 3 operators), the algorithm successfully reproduces the known analytic values for central charge $c$, external weights $h_{ext}$, and internal weights $h_{int}$.
• Truncated Spectra: For models requiring more operators (e.g., 5-state models), the method identifies stable truncated solutions. As the truncation size $N$ increases, the reward $R$ improves until it plateaus. The point where the reward stops improving and the inferred OPE coefficient of the next state becomes statistically consistent with zero (relative to its variance) indicates the correct truncation size.
• Accuracy: The relative errors between found spectra and known minimal models are typically below 0.2%.

B. Candidate Non-Unitary Spectra ($c > 1$)

• Discovery: The authors applied the method to the non-unitary regime with $c > 1$.
• Stability: They identified stable candidate spectra starting at $N=11$ exchanged states.
  • The reward $R$ exceeded the benchmark threshold (established from minimal models) at $N=11$.
  • Increasing to $N=12$ and $N=13$ yielded negligible improvements in $R$.
  • The inferred OPE coefficients for the 12th and 13th states were orders of magnitude smaller than the 11th and exhibited high statistical noise, confirming they were artifacts of truncation rather than physical states.
• Crossing Violation: The crossing violation (residual error) of these $c > 1$ candidates was comparable to that of the known minimal models, suggesting they are valid approximate solutions to the crossing equations.

5. Significance and Outlook

• Beyond Convexity: This work provides a viable "route" for bootstrapping theories where unitarity cannot be assumed, opening the door to studying complex fixed points and logarithmic CFTs.
• Interior Search: The method is uniquely suited for finding specific theories inside the allowed region of CFT data space, rather than just bounding the edges.
• Future Directions:
  • Extension to higher dimensions ($d > 2$) and theories with spin.
  • Incorporation of mixed correlators and modular invariance constraints.
  • Refinement of the truncation error term $e(z, \bar{z})$ using analytic expectations (e.g., light-cone bootstrap asymptotics).
  • Exploration of other optimization strategies (e.g., machine learning) for the non-differentiable objective function.

In summary, the paper establishes a robust numerical framework for solving the conformal bootstrap in non-unitary regimes by leveraging the statistical stability of OPE data, successfully recovering known results and proposing new candidate theories for $c > 1$.