Critical and multicritical Lee-Yang fixed points in the local potential approximation

This paper employs the functional renormalization group in the Local Potential Approximation to trace critical and multicritical Lee-Yang fixed points from their upper critical dimensions down to two dimensions, successfully following the $n=1$ case while revealing that higher-order multicritical fixed points ($n>1$) annihilate with non-perturbative solutions before reaching $d=2$.

The Gist

Imagine you are trying to understand the rules of a game that happens at the very edge of chaos. In physics, this "game" is how materials behave when they are about to change state, like water turning into steam or a magnet losing its magnetism. Scientists call these special moments "critical points," and they are governed by hidden rules called "universality classes."

This paper is a detective story about a very specific, tricky type of game called the Lee-Yang universality class. Here is the simple breakdown of what the authors did, using everyday analogies.

The Mystery: A Game with "Ghost" Rules

Usually, the rules of physics are "real" and straightforward. But the Lee-Yang game is different. It involves a "complex" interaction, which the authors describe as having an imaginary number ($i$) in its equation. Think of this like a game where the dice are made of ghosts.

• The Catch: Even though the rules involve "ghosts" (imaginary numbers), the final results of the game (the patterns you see) are still real and measurable. This is due to a special symmetry called PT symmetry.
• The Goal: The authors wanted to see how this game changes as they shrink the "playground" (the number of dimensions). They started in a high-dimensional playground (6 dimensions) where the rules are easy to calculate, and tried to walk all the way down to a 2-dimensional world (like a flat sheet of paper).

The Tool: The "Zoom Lens" (Functional Renormalization Group)

To study this, the authors used a mathematical tool called the Functional Renormalization Group (FRG).

• The Analogy: Imagine looking at a painting through a zoom lens.
  • When you zoom out (high energy), you see the broad, simple strokes.
  • When you zoom in (low energy), you see the tiny details.
  • The FRG is a way to smoothly zoom from the big picture down to the tiny details without losing the connection between them.
• The Approximation: To make the math solvable, they used a simplified version of the lens called the Local Potential Approximation (LPA). Think of this as looking at the painting through a slightly blurry lens. It's not perfect, but it's the best way to see the whole picture at once. They used two versions: one where the lens is fixed (LPA) and one where the lens can adjust slightly (LPA').

The Journey: Walking from 6D to 2D

The authors tried to trace the "Lee-Yang game" from its starting point in 6 dimensions down to 2 dimensions.

1. The Success Story (The Simple Case):
For the simplest version of the game (called $n=1$), they successfully walked the whole path.

• The Result: They found that the game works all the way down to 2 dimensions.
• The Accuracy: Their "blurry lens" results were surprisingly accurate. When they compared their numbers to the known exact answers for the 2D world, they were off by only a tiny bit (between 2.6% and 7%). It's like guessing the weight of an elephant and being off by just a few pounds.

2. The Problem with the Complex Versions (The Multicritical Cases):
Then they tried to trace more complicated versions of the game (where $n > 1$). These are like harder levels of the same game.

• The Obstacle: As they walked down from 6 dimensions toward 2, they hit a wall.
• The "Ghost" Collision: Around dimension 2.72, something strange happened. New, unexpected "ghost" solutions (fixed points) popped out of nowhere. These new ghosts collided with the original game rules and destroyed them.
• The Conclusion: Because of these collisions, the authors could not continue the complex versions of the game all the way down to 2 dimensions using their current tools. The path simply ended before they reached the finish line.

The Twist: When the Rules Flip

A key discovery in the paper is about a specific number called the scaling dimension (let's call it $\Delta$). This number tells you how "heavy" or "light" the game pieces are.

• In the beginning (6 dimensions), $\Delta$ is positive.
• As they walked down, $\Delta$ got smaller and smaller.
• At a specific point (around dimension 2.72), $\Delta$ hit zero and then turned negative.
• Why this matters: When $\Delta$ turns negative, the math changes completely. It's like the ground suddenly flipping upside down. The authors had to invent a new way to analyze the math to handle this flip, which they did by studying the "shape" of the equations (looking for singularities or "tears" in the math).

The Bottom Line

• What they did: They used a mathematical "zoom lens" to trace a strange, imaginary-number-based physics game from high dimensions down to low dimensions.
• What they found:
  • The simple version of the game works perfectly all the way to 2 dimensions and matches known facts very well.
  • The harder, more complex versions of the game break down before reaching 2 dimensions because they get "eaten" by unexpected new solutions.
• What it means: This suggests that if these complex games do exist in a 2D world, they might not be the simple "imaginary number" games we thought they were. They might need a completely different set of rules that the authors haven't found yet.

In short, the authors successfully mapped the easy path but found a dead end on the hard paths, revealing that the landscape of these physics games is more treacherous and complex than previously thought.

Technical Summary

Technical Summary: Critical and Multicritical Lee-Yang Fixed Points in the Local Potential Approximation

Problem Statement
The paper addresses the challenge of characterizing the non-unitary universality classes associated with the Lee-Yang edge singularity and its multicritical generalizations. These classes arise from scalar field theories with complex $i\phi^{2n+1}$ interactions ($n \in \mathbb{N}$) just below their upper critical dimensions $d_{2n+1} = \frac{4n+2}{2n-1}$. While perturbative methods work near these upper critical dimensions, the theories become strongly coupled as the dimension $d$ is lowered toward $d=2$. In two dimensions, it is conjectured that these fixed points correspond to non-unitary conformal minimal models $M(2, 2n+3)$ or $M(2, 4n+1)$, but a rigorous proof connecting the $i\phi^{2n+1}$ theories to these specific CFTs across dimensions is lacking. Furthermore, the presence of complex couplings and the violation of unitarity (despite $PT$ symmetry) introduce technical difficulties for standard non-perturbative methods like the numerical bootstrap or Monte Carlo simulations.

Methodology
The authors employ the Functional Renormalization Group (FRG) approach, specifically utilizing the Wetterich equation. They work within the Local Potential Approximation (LPA) and the LPA' (which includes wavefunction renormalization and a running anomalous dimension $\eta$). The study focuses on a single scalar field with a complex, $PT$-symmetric potential $v(\phi) = u(\phi) + ih(\phi)$, where $u$ is even and $h$ is odd.

The methodology combines three distinct analytical and numerical strategies:

1. Perturbative $\epsilon$-expansion: The authors derive the fixed-point solutions near the upper critical dimensions $d_{2n+1} - \epsilon$. They generalize previous procedures to the Wetterich equation, expanding the potential in terms of Hermite polynomials to identify the $i\phi^{2n+1}$ fixed points and determine the structure of the anomalous dimension $\eta$.
2. Analytic Properties of Fixed-Point Equations: Before numerical integration, the authors analyze the ordinary differential equations (ODEs) governing the fixed points. They investigate movable singularities, large-field asymptotic behaviors, and the specific case where the scaling dimension $\Delta = (d-2+\eta)/2$ vanishes. This analysis reveals that for $PT$-symmetric potentials, $\eta$ is real but negative, which can lead to $\Delta < 0$ in low dimensions, fundamentally altering the asymptotic behavior of solutions compared to unitary theories.
3. Numerical Shooting: The authors solve the full fixed-point equations numerically. To manage the complexity of the coupled real and imaginary parts, they utilize a hybrid approach: truncating the real part of the potential to a quadratic polynomial (justified by large-field analysis showing the real part is subleading) while solving the imaginary part functionally. They employ a "spike plot" method to identify global solutions (fixed points) by scanning initial conditions and looking for discontinuities where solutions avoid movable singularities.

Key Contributions and Results

• Lee-Yang Universality Class ($n=1$):

  • The authors successfully identify and track the $i\phi^3$ fixed point from its upper critical dimension ($d=6$) down to $d=2$.
  • In the LPA' scheme, they numerically determine the scaling dimension $\Delta_\phi$ of the fundamental field as a function of $d$. The results show good quantitative agreement with exact 2D values, with relative errors between 2.6% and 7%.
  • However, the scaling dimension of the composite operator $\phi^3$ (related to the correction-to-scaling exponent $\omega$) becomes unreliable for $d \lesssim 4$.
  • A critical technical finding is that in the LPA', the scaling dimension $\Delta_\phi$ crosses zero at a specific dimension $d_0 \approx 2.72$. This vanishing of $\Delta$ marks a drastic change in the behavior of the fixed-point equation and the nature of the solutions.

• Multicritical Lee-Yang Fixed Points ($n > 1$):

  • The authors identify fixed points for $n=2$ (corresponding to $i\phi^5$) below their upper critical dimension ($d=10/3$). The result for $\Delta_\phi$ at $d=3$ agrees within 4% with extrapolations from perturbative results and exact 2D data.
  • Crucial Negative Result: Unlike the $n=1$ case, the multicritical fixed points ($n > 1$) cannot be continued to $d=2$ within the LPA' framework. As $d$ decreases, new non-perturbative fixed points branch off from the critical Lee-Yang solution at $d \approx 2.72$. These new branches annihilate the multicritical fixed points one by one at dimensions strictly greater than $d=2$.
  • Consequently, the authors cannot establish a connection between the $i\phi^{2n+1}$ theories and the conjectured minimal models $M(2, 2n+3)$ or $M(2, 4n+1)$ for $n > 1$ using this approximation.

• Analytic Insights:

  • The paper provides a detailed analysis of the dynamical system representation of the fixed-point equations. It demonstrates that for $\Delta < 0$ (which occurs in low dimensions for these models), the origin of the dynamical system becomes stable, allowing for a continuum of global solutions, whereas for $\Delta > 0$, global solutions are isolated.
  • The authors derive an analytical prediction for the dimension $d_0$ where $\Delta=0$ in the LPA' approximation, finding $d_0 \approx 2.7249$, which matches their numerical findings.

Significance and Claims
The paper claims to provide a comprehensive non-perturbative interpolation of the Lee-Yang universality class from $d=6$ to $d=2$ using the FRG, validating the approach for the $n=1$ case while highlighting its limitations for higher multicriticalities.

The authors are modest regarding the interpretation of the failure to reach $d=2$ for $n > 1$. They state that it remains to be understood whether the annihilation of the multicritical fixed points by new non-perturbative branches is an artifact of the LPA' approximation or a genuine non-perturbative feature of the theory. If the latter is true, it would imply that the conjectures linking $i\phi^{2n+1}$ theories to specific minimal models in 2D may require revision, potentially involving versions of the theories that are not $PT$-invariant. The work underscores the necessity of developing higher-order derivative expansions or alternative methods to resolve the fate of multicritical Lee-Yang fixed points in two dimensions.