Long-range minimal models

This paper introduces and analyzes a new class of nonlocal conformal field theories called "long-range minimal models," constructed by deforming Virasoro minimal models with generalized free fields, and investigates their perturbative and nonperturbative properties, particularly highlighting the challenges at large $m$ for $\phi_{2,2}$ deformations versus the well-behaved large-$m$ limits and new Mellin amplitude techniques for $\phi_{1,2}$ deformations.

The Gist

Imagine the universe of physics as a giant, complex dance floor. Usually, dancers (particles) only interact with the people standing right next to them. This is how most of our standard physics works: locality. If you want to change the dance, you push your neighbor, who pushes their neighbor, and so on.

But what if dancers could reach across the entire room to grab hands with people on the other side? This is the world of Long-Range Models. In these systems, interactions happen over vast distances, like a magical connection that doesn't fade quickly with distance.

This paper is about exploring a specific, exotic version of this "long-distance dance" in a two-dimensional world. The authors are trying to understand what happens when you take a very well-known, perfectly choreographed dance (called a Minimal Model) and introduce a "long-range" partner to dance with it.

Here is the breakdown of their journey, using simple analogies:

1. The Setup: Mixing Two Types of Dancers

The authors start with a classic, local dance troupe (the Virasoro Minimal Models). These are famous in physics because they are "solvable"—we know exactly how they move.

• The Twist: They introduce a new, invisible partner called a Generalized Free Field (GFF). Think of this GFF as a ghostly dancer who can instantly connect with anyone in the room, regardless of distance.
• The Interaction: They couple the local troupe to this ghostly partner. The result is a new, hybrid dance troupe. Because of the ghostly partner, the whole system becomes non-local. The dancers are no longer just reacting to their neighbors; they are reacting to the whole room.

2. The Goal: Finding the "Sweet Spot" (Fixed Points)

When you change the rules of a dance (by turning on this long-range connection), the system tries to find a new rhythm.

• The Flow: As the system settles down (flows to the "infrared"), it reaches a new, stable rhythm called a Fixed Point.
• The Discovery: The authors found a whole new family of these stable rhythms, which they call Long-Range Minimal Models (LRMMs). These are new types of universes with their own unique rules of physics.

3. The Challenge: The "Goldilocks" Problem

The authors tried to study these new models using two different mathematical tools (perturbative expansions), which are like two different ways of estimating the dance moves:

• Tool A (Near Mean Field): Good for when the long-range connection is very weak.
• Tool B (Near Short-Range): Good for when the connection is just about to break and become local again.

The Problem: They found that for many of these models, both tools fail when the "size" of the system (represented by a number called $m$) gets too big. It's like trying to use a ruler to measure the distance to the moon; the tool just isn't built for that scale.

• The Consequence: To understand the middle ground (where $m$ is large), they needed to invent new, "non-perturbative" methods. They couldn't just guess; they had to calculate the exact moves.

4. The Success Stories: Two Different Families

The paper focuses on three specific types of these hybrid dances. Two of them turned out to be very tricky, but one was surprisingly well-behaved.

• The Tricky Ones (The $\phi_{2,2}$ and $\phi_{2,1}$ families):
  These models are like a chaotic dance floor where the rules change wildly as the system gets bigger. The authors had to use heavy-duty math (involving complex integrals and computer code) to figure out the behavior. They found that as the system grows, the math gets messy, and the "long-range" nature makes it hard to predict the outcome using simple approximations.

• The Well-Behaved One (The $\phi_{1,2}$ family):
  This is the star of the show. Even though it looks similar to the tricky ones, it behaves beautifully when the system gets large.

  • The Analogy: Imagine a chaotic jazz band (the tricky models) vs. a perfectly synchronized marching band (the $\phi_{1,2}$ model). The marching band follows a simple, predictable pattern even as the band gets huge.
  • The Breakthrough: The authors proved this using a clever mathematical trick involving Mellin Amplitudes. Think of this as translating the dance moves from "English" (complicated integrals) into "Morse Code" (a simpler mathematical language). This translation revealed that the complex math actually hides a very simple, elegant formula.

5. The "Magic Translator" (Coulomb Gas & Mellin Space)

How did they crack the code for the well-behaved model?

• They used a method called the Coulomb Gas formalism. Imagine the dance floor is filled with charged particles. Instead of tracking every single step, they looked at the "electric field" created by the whole group.
• They then used Mellin Space. This is like taking a photo of the dance and turning it into a spectrum of colors. Instead of seeing the messy dance moves, they saw a clean, simple pattern of colors that revealed the underlying rules.
• The Result: They found that for this specific model, the "anomalous dimensions" (a fancy physics term for how the dancers' energy changes) follow a simple, predictable formula that works perfectly for huge systems.

Summary: Why Does This Matter?

This paper is a map for exploring a new territory in physics.

1. New Universes: It proves that there are entire families of "Long-Range" universes that we didn't fully understand before.
2. New Tools: It shows that when standard math fails (because things get too big), we can use "translation" techniques (like Mellin space) to find simple answers hidden inside complex problems.
3. The Future: It opens the door to studying other complex systems, like materials with long-range interactions or even models of disorder in nature, using these new "Long-Range Minimal Models."

In short, the authors took a complex, non-local physics problem, realized their standard tools broke down, invented a new "translator" to speak the language of the problem, and discovered that one specific version of this problem is actually beautifully simple and predictable.

Technical Summary

1. Problem Statement and Motivation

The paper investigates a class of nonlocal Conformal Field Theories (CFTs) in two dimensions, termed Long-Range Minimal Models (LRMMs). These models arise from deforming unitary, diagonal Virasoro minimal models ($M_{m+1,m}$) by coupling a relevant primary operator $\phi_{r,s}$ to a Generalized Free Field (GFF), denoted as $\chi$.

• Context: While 2D local CFTs are well-understood via the infinite-dimensional Virasoro algebra, nonlocal CFTs (lacking a local energy-momentum tensor) are less explored. They naturally appear in statistical mechanics as long-range interacting models (e.g., spins interacting with power-law decay).
• Construction: The authors define the theory by deforming the expectation values of the minimal model with an interaction term $g_0 \int d^2x \, \phi_{r,s} \chi$. The GFF $\chi$ has a scaling dimension $\Delta_\chi = 2 - \Delta_{r,s} - \delta$ (where $0 < \delta \ll 1$), making the composite operator $O = \phi_{r,s}\chi$ weakly relevant ($\Delta_O = 2-\delta$).
• Goal: To characterize the infrared (IR) fixed points of these flows, compute their critical exponents (beta functions and anomalous dimensions), and understand the behavior of these models in the large-$m$ limit (where $m$ labels the minimal model).

2. Methodology

The authors employ a multi-faceted approach combining conformal perturbation theory, numerical analysis, and advanced analytic techniques:

1. Conformal Perturbation Theory (CPT):

   • The beta function $\beta(g)$ and anomalous dimensions $\gamma(g)$ are computed using standard CPT.
   • The beta function is derived from the integral of the four-point function of the perturbing operator $O$:
     $$\beta_3 = -\pi \int_{\mathcal{R}} d^2z \, \langle O(0)O(z,\bar{z})O(1)O(\infty) \rangle_{\text{finite}}$$
   • Anomalous dimensions are calculated via the renormalization of two-point functions involving Virasoro primaries and higher-spin currents.

2. Numerical Integration and Extrapolation:

   • For finite $m$, the authors compute the required integrals numerically. They utilize the BPZ differential equations to expand Virasoro blocks (conformal blocks) in radial coordinates ($\rho$).
   • They truncate the series expansion of the blocks to a high order ($n_{\text{max}}$) to ensure convergence and stability.
   • Numerical data for various $m$ (up to $m \approx 20$) are fitted to polynomial expansions in $1/m$ to conjecture large-$m$ behavior.

3. Analytic Large-$m$ Approaches:

   • Multi-coupling RG Flow: For the $(m, 1, 2)$ case, they introduce a multi-coupling flow involving $\phi_{1,2}\chi$ and $\phi_{1,3}$ to handle UV divergences that appear when taking $m \to \infty$ before integration.
   • Mellin-Barnes (MB) Representation & Coulomb Gas: To prove the large-$m$ results analytically, they utilize the Coulomb gas formalism to represent correlators as integrals. They then convert these into Mellin-Barnes integrals.
   • Contour Deformation: A key technical innovation is the systematic deformation of MB contours to avoid "pinching" singularities as parameters approach the large-$m$ limit. This is implemented using the Mathematica package MB, allowing for the extraction of analytic expressions for coefficients that were previously only known numerically.

3. Key Contributions and Results

The paper focuses on three specific families of LRMMs: $(m, 2, 2)$, $(m, 1, 2)$, and $(m, 2, 1)$.

A. The $(m, 2, 2)$ Family (Generalized Long-Range Ising)

• Physical Interpretation: These models are conjectured to be the long-range deformations of the multicritical Landau-Ginzburg theories (generalizing the Ising model, $m=3$).
• Beta Function: They computed $\beta_3$ numerically for many $m$ and found it is positive, ensuring unitary fixed points.
  • Large-$m$ Limit: $\beta_3 \sim \frac{\pi^2}{2m^2} - \frac{\pi^2}{2m^3} + O(m^{-4})$.
• Anomalous Dimensions:
  • For Virasoro primaries $\phi_{r,s}$, they derived a conjectured formula for the leading large-$m$ behavior:
    $$\gamma_{r,s} \sim \pm \frac{m}{2} \frac{(r-s)^2(r+s)}{(r+1)(s-1)} \delta$$
    (Sign depends on whether $r \le s$ or $r > s$).
  • Higher-Spin Currents: They computed the anomalous dimensions of broken higher-spin currents (e.g., stress tensor $T$, spin-4, spin-6) using the multiplet recombination method.
• Problematic Large-$m$ Limit: The authors found that the perturbative expansion breaks down for large $m$ in this regime because the acceptable range for the perturbation parameter $\epsilon$ shrinks exponentially. Non-perturbative methods are required in the intermediate range.

B. The $(m, 1, 2)$ and $(m, 2, 1)$ Families

• Duality: These models are related by a symmetry exchanging Kac labels $(r, s) \leftrightarrow (s, r)$ and $m \to -m$.
• Mixing Problem: A novel feature in the $(m, 1, 2)$ case is operator mixing. For odd $r$, the operator $\phi_{r,1}$ mixes with quasiprimary operators involving the GFF $\chi$ (e.g., $\chi \phi_{r,2}$). This leads to non-vanishing one-loop contributions to anomalous dimensions, unlike the $(m, 2, 2)$ case.
• Well-Behaved Large-$m$ Limit: Unlike the $(m, 2, 2)$ case, the $(m, 1, 2)$ family exhibits a well-behaved large-$m$ limit.
  • Beta Function: $\beta_3 \sim \frac{3\pi^2}{8}(m - 1 - 8\log 2) + O(m^{-1})$.
  • Anomalous Dimensions: They derived a closed-form expression for the leading order:
    $$\gamma_{r,s} \sim \frac{1}{3}(r-s)^2 \frac{s + (-1)^{s+r+1}}{s + (-1)^{s+r}} \delta$$
• Non-Unitary Case: The $(m, 2, 1)$ model yields a complex fixed point (negative $\beta_3$), indicating non-unitarity, consistent with the symmetry arguments.

C. Analytic Verification via Mellin Amplitudes

• The authors successfully used the Mellin-Barnes/Coulomb gas method to prove the conjectured large-$m$ formulas for the $(m, 1, 2)$ and $(m, 2, 2)$ cases.
• This method revealed analytic expressions for integrals in conformal perturbation theory that were previously only accessible via numerical integration.
• They demonstrated that for specific operators (like $\phi_{r,1}$ with even $r$ in the $(m, 1, 2)$ flow), the anomalous dimensions vanish, generalizing known results for the 2D Ising model.

4. Significance and Future Directions

• New Class of CFTs: The paper establishes a systematic framework for constructing and analyzing nonlocal CFTs based on minimal models, extending the known long-range Ising results to multicritical systems.
• Technical Breakthrough: The application of Mellin-Barnes techniques to minimal model correlators provides a powerful new tool for extracting large-$N$ (or large-$m$) limits in CFTs, bypassing the need for brute-force numerical fitting.
• Duality and Crossover: The work clarifies the duality between the long-range multicritical models and the short-range minimal models perturbed by a GFF, particularly near the crossover point where the theory transitions from long-range to short-range universality classes.
• Future Applications:
  • The framework can be extended to non-unitary minimal models and long-range Potts models.
  • The results provide crucial data for the conformal bootstrap to test bounds on long-range theories.
  • The study of defect operators (pinning fields) in these nonlocal theories is proposed as a future direction.

In summary, this paper provides a comprehensive perturbative and analytic study of long-range deformations of 2D minimal models, uncovering distinct behaviors in different sectors, resolving large-$m$ limits analytically, and offering a robust toolkit for future investigations into nonlocal CFTs.