---
title: Long-Range Minimal Models in 2D CFT
url: https://www.emergentmind.com/topics/long-range-minimal-models
type: topic
---

# Long-Range Minimal Models in 2D CFT

Searching arXiv for recent papers directly related to long-range minimal models and adjacent constructions.
Long-range minimal models are a class of nonlocal conformal field theories in two dimensions obtained as deformations of Virasoro minimal models by coupling a relevant primary $\phi_{r,s}$ to a generalized free field $\chi$ so that the composite operator $O(x)=\phi_{r,s}(x)\chi(x)$ has scaling dimension $2-\delta$; the deformation
$$
S_{\rm int}=g_0\int d^2x\,O(x)=g_0\int d^2x\,\phi_{r,s}\chi
$$
can then flow to an infrared fixed point at $g_*^2=\delta/\beta_3$ when the cubic coefficient of the beta function is real and positive [2509.26372]. In this usage, the long-range minimal model (LRMM) of type $(m,r,s)$ is the infrared fixed point associated with the unitary, diagonal Virasoro minimal model $\mathcal M_{m+1,m}$ and a bosonic generalized free field chosen so that the perturbation is weakly relevant [2509.26372].

## 1. Definition as deformations of Virasoro minimal models

The construction starts from the unitary, diagonal Virasoro minimal model $\mathcal M_{m+1,m}$, $m\ge 3$, with central charge
$$
c=1-\frac{6}{m(m+1)},
$$
and primary fields $\phi_{r,s}$ of weight
$$
h_{r,s}=\frac{[((m+1)r-ms)^2-1]}{4m(m+1)}.
$$
A bosonic generalized free field $\chi(x)$ is then introduced with scaling dimension
$$
\Delta_\chi=2-\Delta_{r,s}-\delta,\qquad 0<\delta\ll 1,
$$
so that
$$
\Delta_O=\Delta_{r,s}+\Delta_\chi=2-\delta.
$$
The ultraviolet theory at $g_0=0$ is therefore the nonlocal CFT given by “minimal model $\boxtimes$ GFF,” deformed by $O=\phi_{r,s}\chi$ [2509.26372].

Under the assumptions stated for standard conformal perturbation theory—specifically, that no relevant or marginal composite operators mix with $O$, provided $\delta\neq 0$ and no other OPE channel is near marginality—the beta function truncates to
$$
\beta(g)=-\delta\,g+\beta_3\,g^3+O(g^5).
$$
A real zero at
$$
g_*^2=\frac{\delta}{\beta_3}
$$
defines the infrared fixed point. This criterion is the operational definition of the LRMM in the 2025 construction [2509.26372].

The construction is explicitly nonlocal because the generalized free field sector is not generated by a local two-dimensional CFT. A plausible implication is that the adjective “long-range” refers not to a lattice interaction written directly in real space, but to the nonlocal continuum sector that is coupled to the minimal-model primary.

## 2. The $(m,2,2)$ family and the two perturbative regimes

The family based on $O=\phi_{2,2}\chi$ is singled out because it admits two complementary weak-coupling descriptions. In this case,
$$
\Delta_{2,2}=\frac{3}{2m(m+1)},\qquad \Delta_\chi=2-\Delta_{2,2}-\delta.
$$
The first description is a near-mean-field regime. One begins with a generalized free field $\phi$ in $d=2$ and adds a local $2(m-1)$-point vertex. Equivalently, for $\delta\ll 1$ one chooses
$$
s=\bar s+\frac{\delta}{m-1},\qquad \bar s=\frac{2(m-2)}{m-1},
$$
which gives a weakly relevant coupling $\lambda_0$ for $\phi^{2(m-1)}$. The renormalized beta function is
$$
\beta_{\rm MFT}(\lambda)=R\frac{d\lambda}{dR}=-\lambda+A\lambda^2-2B_1\lambda^3+O(\lambda^4),
$$
with
$$
A=\pi\,\frac{(2m-2)!}{(m-1)!^3}.
$$
The infrared fixed point obeys $\lambda_*\simeq A^{-1}+O(\delta)$, and the weak anomalous dimension of the interaction operator is
$$
\Delta_{\phi^{2(m-1)}}=2+\beta'(\lambda_*)=2+\delta-\frac{2B_1\delta^2}{A^2}+O(\delta^3)
$$
[2509.26372].

The second description is a near-short-range, or Sak, regime obtained by conformal perturbation around $\mathcal M_{m+1,m}\boxtimes{\rm GFF}$. One turns on
$$
S_{\rm int}=g\int \phi_{2,2}\chi,
$$
and finds again
$$
\beta(g)=-\delta g+\beta_3 g^3+O(g^5),
$$
with
$$
\beta_3=-\pi\int_{\mathcal R} d^2z\,\langle O(0)O(z)O(1)O(\infty)\rangle_0\big|_{\rm finite}.
$$
For integer $m$, the integral is evaluated by radial and block expansions, and $\beta_3$ is positive for all $m$. Its large-$m$ asymptotic is
$$
\beta_3=\frac{\pi^2}{2m^2}-\frac{\pi^2}{2m^3}+O(m^{-4}).
$$
The leading anomalous dimensions of Virasoro primaries satisfy
$$
\gamma_{r,s}(g)=-\frac12\,B_{r,s}\,g^2+O(g^4),
$$
while higher-spin currents in the identity multiplet acquire nonzero anomalous dimensions, for example
$$
\gamma_2=(\pi g_*)^2\,c^{-1}\Delta_{2,2}\Delta_\chi.
$$
These two expansions are proposed to be dual descriptions of the same line of fixed points, with the shadow relation
$$
\Delta_{2,2}+\Delta_\chi=2
$$
as the basic matching condition [2509.26372].

The large-$m$ limit is problematic in both perturbative regimes. The stated consequence is that nonperturbative methods are required in the intermediate range for all values of $m$ [2509.26372].

## 3. Models based on $\phi_{1,2}$

The families based on $O=\phi_{1,2}\chi$ and $O=\phi_{2,1}\chi$ differ qualitatively from the $(m,2,2)$ case. Here there is only one known weakly relevant perturbative expansion, and it is reported to be well behaved at large $m$. The same conformal-perturbative method applies, but the OPE now contains only two blocks [2509.26372].

At one loop, the beta function again has the form
$$
\beta(g)=-\delta\,g+\beta_3\,g^3+\cdots,
$$
with $\beta_3$ extracted from the same basic four-point integral. Numerically, for finite $m$, one finds $\beta_3>0$, while the large-$m$ asymptotic is
$$
\beta_3=\frac{3\pi^2}{8}(m-1-8\ln 2)+O(m^0).
$$
The leading scaling dimensions obey
$$
\gamma_{r,s}(g)=-\frac12\,B_{r,s}\,g^2,
$$
and, for operators that do not mix, one has at $g=g_*$
$$
\gamma_{r,s}(g_*)\simeq \frac13\,(r-s)^2\,[\,s+(( -1)^{r+s+1}s)+(( -1)^{r+s})(\delta)\,]+O(m^{-1}).
$$
The construction also exhibits operator mixing: $\phi_{r,1}$ with odd $r$ mix with $\chi\,\phi_{r,2}$, producing one-loop splittings; a representative example is
$$
\gamma(g)=\pm \frac{m+2}{m+3}(\pi g)+O(g^2)
$$
for $\phi_{3,1}$ [2509.26372].

A systematic two-loop computation for infinitely many primaries $\phi_{r,s}$ in type $(m,1,2)$ was carried out, and the large-$m$ limits were obtained both numerically and through Mellin-amplitude methods. In the terminology of the source, this confirms that the $\phi_{1,2}$ family is comparatively well behaved at large $m$ [2509.26372].

## 4. Coulomb-gas and Mellin-Barnes techniques

A central technical issue is that standard large-$m$ perturbation theory around minimal models becomes difficult because $\beta_3$ and $B_{r,s}$ grow rapidly with $m$. To address this, the construction introduces a “chiral first” Coulomb-gas approach together with Mellin-Barnes methods [2509.26372].

In the chiral approach, one expands in $m\to\infty$ before integrating, using the Mellin representation
$$
(t-u)^{-\alpha}=\int \frac{d\gamma}{2\pi i}\,\Gamma(-\gamma)\Gamma(\alpha+\gamma)t^\gamma(-u)^{-\alpha-\gamma}.
$$
Contour deformation then yields explicit large-$m$ expansions of Virasoro blocks. The first two examples quoted are
$$
\mathcal F_{1,1}(z)= \frac{2-z}{2\sqrt{1-z}} + \frac{3z\ln(1-z)}{4m\sqrt{1-z}}+\cdots,
$$
and
$$
\mathcal F_{1,3}(z)= \frac{z}{\sqrt{1-z}} - \frac{(z+2)\ln(1-z)+2z(3-4\ln z)}{2m\sqrt{1-z}}+\cdots.
$$
For the full four-point function needed in $\beta_3$, two successive Symanzik star integrals lead to a seven-fold Mellin integral. The analytic continuations are then performed with the Mathematica package MB, and the resulting expansion reproduces
$$
\beta_3=\frac{\pi^2}{2m^2}-\frac{\pi^2}{2m^3}+O(m^{-4})
$$
for the $\phi_{2,2}\chi$ family [2509.26372].

The stated significance of these methods is twofold. First, they provide analytic access to integrals that had previously been known only numerically. Second, they produce large-$m$ asymptotics for infinitely many anomalous dimensions, especially in the $\phi_{1,2}$ family, where the perturbative expansion remains controlled [2509.26372].

## 5. Relation to long-range deformations of non-unitary minimal series

A distinct but closely related line of work studies long-range deformations of the non-unitary minimal series $\mathcal M(2,2m+1)$ through a nonlocal kinetic term with propagator $\langle \phi(p)\phi(-p)\rangle_0\sim 1/|p|^{2\sigma}$ and through deformations of the short-range minimal model by a long-range field $\chi$ [2603.27031]. In that setting, two constructions are compared: a Landau-Ginzburg action with interaction $i g_0\phi^{2m-1}/(2m-1)!$ and a direct deformation
$$
S_{\rm MF}=S_{\mathcal M(2,2m+1)}+S_0[\chi]+g_0\int d^2x\,\phi_{1,2}(x)\chi(x).
$$
The perturbative result is sharply different from the unitary LRMM story: for $m=2$ one finds $\beta_3>0$, real nontrivial fixed points, and a spectrum match between the two constructions, whereas for every $m>2$ one finds $\beta_3<0$, so $g_*^2$ is negative, $g_*$ is pure imaginary, and the stress tensor becomes slightly relevant [2603.27031].

The $m=2$ case is the long-range Lee-Yang model. In this special case, the source states that the long-range Lee-Yang fixed point coincides with the known long-range Ising universality class after appropriate analytic continuation of exponents, and both flows produce the same two low-lying operator dimensions and the same marginality crossover [2603.27031]. Independent support for this broader perspective comes from one-dimensional bootstrap sum rules: “Super Sum rules for Long-Range Models” tests long-range versions of the Ising, $O(N)$, and Lee-Yang models and finds that the sum rules correctly predict the CFT data characterising these theories, while also substantially reducing the allowed parameter space in numerical bootstrap studies [2603.22395].

A plausible implication is that “long-range minimal models” is not a uniformly stable category across minimal-series constructions: its viability depends strongly on which minimal series is deformed and on whether one works in a unitary or non-unitary setting.

## 6. Open problems, crossover structure, and neighboring meanings

Several unresolved issues define the current state of the subject. In the unitary LRMM construction, the $(m,2,2)$ family has two complementary perturbative descriptions but a problematic large-$m$ limit in both regimes, so nonperturbative methods are required in the intermediate range for all values of $m$. In the non-unitary $\mathcal M(2,2m+1)$ constructions, the short-range limit $s\to 2$ is described as subtle because perturbation theory around nonlocality breaks down as wave-function renormalization develops poles at $s=2$; functional RG or Hamiltonian truncation is identified as needed to connect the long- and short-range regimes smoothly [2509.26372; 2603.27031].

The subject also sits near, but is not identical to, other long-range theories in mathematical physics. In long-range Ising models on $\mathbb Z^d$, for instance, there are rigorous results on planelike interfaces, class A minimal sets, and the reciprocal approximation of lattice ground states and nonlocal minimal surfaces. In that literature, if $K(x,y)$ satisfies
$$
\lambda |x-y|^{-(d+s)}\le K(x,y)\le \Lambda |x-y|^{-(d+s)},\qquad s\in(0,1),
$$
one obtains nonlocal perimeter minimizers whose boundaries stay in a strip of bounded width, together with $T$-convergence from discrete long-range Ising energies to continuum nonlocal perimeter functionals [1605.06187]. These objects are mathematically “minimal,” but they are not minimal models in the Virasoro-CFT sense.

Within conformal field theory, therefore, long-range minimal models designate a specific program: constructing nonlocal two-dimensional fixed points by coupling minimal-model primaries to generalized free fields, analyzing their beta functions and anomalous dimensions, and comparing complementary long-range and short-range descriptions. The current picture is most developed for the unitary $(m,r,s)$ construction and for the special Lee-Yang case among non-unitary series, while the general nonperturbative structure remains open [2509.26372].

Source: https://www.emergentmind.com/topics/long-range-minimal-models