The Lee-Yang model and its generalizations through the lens of long-range deformations

This paper investigates the conjecture that non-unitary conformal minimal models arise from complex scalar field theories by testing long-range deformations, finding inconsistencies for $m>2$ but confirming that the long-range Lee-Yang model ($m=2$) behaves analogously to the long-range Ising model.

The Gist

Imagine the universe of physics as a giant, complex video game. In this game, there are different "levels" or "worlds" that describe how particles interact. Some worlds are stable and follow the usual rules of cause and effect (like our everyday reality). Others are weird, unstable, and involve "ghostly" rules where things can have negative energy or imaginary numbers.

This paper is about investigating one of these weird, ghostly worlds called the Lee-Yang model and its more complex cousins. The author, Fanny Eustachon, is trying to figure out if two different ways of building this world actually lead to the same place, or if they are actually two different worlds that just look similar from a distance.

Here is the story of the paper, broken down into simple concepts:

1. The Two Blueprints

Imagine you want to build a house (a physical theory). You have two different blueprints:

• Blueprint A (The "Imaginary" House): You start with a standard, empty house and add a very strange, "imaginary" ingredient to the walls. In math, this is like adding a term $i\phi^3$ (or higher powers) to the recipe. This recipe is known to create the Lee-Yang world.
• Blueprint B (The "Long-Range" House): You start with a finished, complex house (a "Minimal Model" from a famous list of perfect mathematical structures) and connect it to a long, stretchy rubber band that reaches across the universe. This rubber band represents "long-range" interactions.

The big question is: If you build the house using Blueprint A and the house using Blueprint B, do you end up with the exact same house?

2. The "Magic Knob" (The Parameter $s$)

To test this, the author introduces a "magic knob" called $s$.

• When you turn the knob to one setting, the interactions are short (like neighbors talking to each other).
• When you turn the knob to another setting, the interactions become "long-range" (like neighbors shouting across the whole city).

The author turns this knob and watches what happens to the house as it settles down into its final form (what physicists call a "fixed point").

3. The Success Story: The Lee-Yang Model ($m=2$)

First, the author tests the simplest version of this weird world, the Lee-Yang model.

• The Result: It works perfectly!
• The Analogy: Whether you build it using the "Imaginary Ingredient" recipe or the "Rubber Band" connection, you end up with the exact same house. The math checks out, the rooms line up, and the energy levels match. It's like two different chefs making a cake, and when they taste it, it's identical. This confirms that for this specific model, the two blueprints are actually the same thing.

4. The Failure Story: The Complex Cousins ($m > 2$)

Next, the author tries to build the more complex, "multicritical" versions of these worlds (where the interaction is $i\phi^5$, $i\phi^7$, etc.).

• The Result: The blueprints do not match.
• The Analogy:
  • Blueprint A builds a house that is weird but stable (it has "complex" numbers, but the physics still makes sense in a specific way).
  • Blueprint B tries to build the same house, but the rubber band gets tangled. The house becomes unstable. It's like trying to build a skyscraper on a foundation that is sinking; the structure collapses or turns into something that doesn't exist in reality (mathematically, the "fixed point" becomes complex in a way that breaks the rules of the game).

The author found a specific "glitch": In the complex versions, the "Rubber Band" blueprint creates a situation where the energy of the house can drop infinitely low (a "sign flip"). In physics, this means the universe would collapse instantly. Therefore, the two blueprints cannot be describing the same reality for these complex models.

5. The Mystery of the "Edge" ($s \to 2$)

There is one final puzzle. The author notes that as the "magic knob" is turned to a specific limit (where the long-range interactions become short-range), the math gets very messy. It's like trying to walk through a foggy wall; the equations break down, and we don't fully understand what happens right at that boundary. The author suspects that "ghostly" non-perturbative effects (things we can't see with our current math tools) are happening there, hiding the true nature of the connection.

Summary

• The Goal: To see if two different mathematical ways of describing "ghostly" physics are actually the same.
• The Good News: For the simplest "ghostly" model (Lee-Yang), the two methods do match. They are twins.
• The Bad News: For the more complex "ghostly" models, the two methods do not match. One builds a stable (though weird) world, while the other builds a collapsing, unstable one.
• The Takeaway: The idea that all these complex models can be described by a simple "imaginary interaction" recipe is likely false for the complex cases. The universe is more complicated than we hoped, and the two blueprints are not interchangeable.

In short: The author checked the math, found a match for the simple case, but discovered a fatal flaw in the complex cases, proving that the "long-range" and "imaginary interaction" descriptions are not the same thing for the more complex models.

Technical Summary

1. Problem Statement

The paper investigates the validity of a recent conjecture regarding the Lee-Yang multicritical class of non-unitary Conformal Field Theories (CFTs), denoted as $M(2, 2m+1)$ for $m \in \mathbb{N}, m \geq 2$.

• The Conjecture: These minimal models are hypothesized to be the Renormalization Group (RG) fixed points of scalar field theories with a purely imaginary interaction term, $i\phi^{2m-1}$, in two dimensions ($d=2$).
• The Challenge: While the $m=2$ case (the standard Lee-Yang model, $M(2,5)$) is well-understood and linked to the $i\phi^3$ theory, the generalization to $m > 2$ is debated. Direct perturbative studies in $d=2$ are difficult due to strong coupling and the non-unitary nature of the theories (complex OPE coefficients, violation of unitary bounds), which hinders standard non-perturbative techniques like the conformal bootstrap.
• The Goal: The author tests the consistency of this conjecture by constructing long-range deformations of both the Landau-Ginzburg ($i\phi^{2m-1}$) formalism and the minimal model ($M(2, 2m+1)$) directly in $d=2$. The study aims to determine if these two distinct constructions yield dual descriptions of the same infrared (IR) fixed point.

2. Methodology

The paper employs Conformal Perturbation Theory (CPT) within a Long-Range (LR) framework. The methodology involves two parallel constructions:

A. Construction 1: Deformation of the Landau-Ginzburg Formalism

• Setup: A Generalized Free Field (GFF) $\phi$ in $d=2$ is defined with a non-local kinetic term characterized by a tunable parameter $s$. The scaling dimension of the field is $\Delta_\phi = 1 - s/2$.
• Deformation: The action is perturbed by a local, purely imaginary interaction:
  $$S_{LR} = S_{GFF} + i\lambda_0 \int d^2x \, \phi^{2m-1}(x)$$
• Analysis: The beta function $\beta(\lambda)$ is computed near the mean-field theory (MFT) regime (where the interaction is nearly marginal). The fixed points are identified, and the anomalous dimensions of composite operators are calculated.

B. Construction 2: Deformation of the Minimal Model

• Setup: The short-range minimal model $M(2, 2m+1)$ is coupled to an independent GFF $\chi$ via a linear deformation involving the primary operator $\phi_{1,2}$ (the field with the smallest conformal dimension in absolute value).
• Deformation:
  $$S'_{LR} = S_{M(2, 2m+1)} + S_{GFF}[\chi] + g_0 \int d^2x \, \phi_{1,2}\chi$$
• Analysis: The RG flow is studied for $s < s^*$ (where $s^*$ is a crossover value). The beta function for the coupling $g$ is computed using four-point functions involving the unperturbed theory (GFF $\times$ Minimal Model). The consistency of the resulting IR fixed point is checked against the shadow relations and the spectrum of the first construction.

C. Consistency Checks

The author compares the two constructions by checking:

1. Fixed Point Reality: Do both constructions yield real or complex conjugate fixed points?
2. Sign Consistency: Does the effective non-local kinetic term generated by integrating out the auxiliary field have the correct sign (ensuring a bounded action)?
3. Spectrum Continuity: Do the scaling dimensions of operators (specifically the stress-energy tensor $T$) match the expectations for a stable IR fixed point?

3. Key Results

Case $m = 2$ (The Lee-Yang Model)

• Consistency: The two constructions are consistent.
  • The $i\phi^3$ deformation yields a pair of complex conjugate fixed points (purely imaginary coupling), which is characteristic of non-unitary real Euclidean QFTs.
  • The $M(2,5)$ deformation yields real fixed points for the coupling $g$.
  • The scaling dimensions match: The operator $\phi^2$ in the $i\phi^3$ theory corresponds to the "shadow" of $\phi$, while in the minimal model construction, the operator $\phi_{1,2}$ couples to $\chi$ such that their dimensions satisfy the shadow relation $\Delta_{\phi_{1,2}} = 2 - \Delta_\chi$.
  • The stress-energy tensor remains irrelevant in the IR, preserving stability.
• Conclusion: The long-range Lee-Yang model is a valid analogue to the long-range Ising model, confirming the conjecture for $m=2$.

Case $m > 2$ (Multicritical Models)

• Inconsistency: The two constructions fail to match, leading to contradictions.
  1. Complex Fixed Points in Minimal Model: For $m > 2$, the beta function for the minimal model deformation yields a coefficient $\beta_3 < 0$. This results in a pair of complex fixed points ($g^2 \propto -\delta$).
  2. Sign Flip: Integrating out the auxiliary field $\chi$ in the minimal model construction generates a non-local kinetic term with a wrong sign (negative). This implies the energy is unbounded from below, rendering the theory unstable.
  3. Stress-Energy Tensor Instability: In the IR, the scaling dimension of the stress-energy tensor $T$ drops below the unitary bound ($\Delta_T < 2$) and becomes slightly relevant ($\gamma_T < 0$). This indicates the fixed point is unstable and flows away, rather than representing a stable CFT.
  4. Divergence from Landau-Ginzburg: While the $i\phi^{2m-1}$ theory still possesses complex conjugate fixed points (consistent with non-unitary QFTs), the minimal model side does not produce a dual description. The "shadow" relation and spectrum continuity break down.

4. Significance and Implications

• Refutation of Generalization: The paper provides strong evidence that the conjecture identifying $M(2, 2m+1)$ with $i\phi^{2m-1}$ theories fails for $m > 2$ in two dimensions. The duality that holds for the Lee-Yang model ($m=2$) does not extend to higher multicritical points.
• Nature of Non-Unitary Fixed Points: The results highlight the fragility of non-unitary fixed points in $d=2$. The transition from $m=2$ to $m>2$ involves a qualitative change where the fixed points become unstable or the duality breaks due to the sign of the OPE coefficients (specifically the $\phi_{1,2} \times \phi_{1,2} \to \phi_{1,3}$ channel).
• Limit $s \to 2$: The paper notes a puzzle regarding the limit $s \to 2$ (recovering the local theory). Perturbative results at finite $s$ cannot be straightforwardly continued to $s=2$ due to poles in the wavefunction renormalization. This suggests that non-perturbative effects are crucial in the region $s > 2$, potentially explaining why the fixed points "annihilate" or become unstable as one approaches the local multicritical models.
• Methodological Contribution: The work demonstrates the utility of long-range deformations as a probe for the consistency of CFT conjectures. By introducing the tunable parameter $s$, the author could access weak-coupling regimes to perform perturbative checks that are impossible in the strictly local $d=2$ theory.

Conclusion

The paper concludes that while the Lee-Yang model ($m=2$) is successfully described as a long-range deformation of both the $i\phi^3$ theory and the $M(2,5)$ minimal model, this duality breaks down for all $m > 2$. The inconsistencies found in the RG flow of the minimal model construction (unstable fixed points, wrong sign kinetic terms) suggest that the proposed Landau-Ginzburg description for the general $M(2, 2m+1)$ class is likely incorrect or requires significant modification beyond the simple $i\phi^{2m-1}$ interaction.