---
title: Long-Range Lee-Yang Model
url: https://www.emergentmind.com/topics/long-range-lee-yang-model
type: topic
---

# Long-Range Lee-Yang Model

The long-range Lee-Yang model is the \(m=2\) member of a family of long-range deformations associated with the non-unitary minimal models \(\mathcal{M}(2,2m+1)\). In the formulation analyzed in 2026, it is studied through two separate long-range constructions: a generalized free scalar field deformed by an imaginary cubic interaction, and a deformation built from the two-dimensional Lee-Yang minimal model \(\mathcal{M}(2,5)\) coupled to a nonlocal generalized free field. The central result is that, in contrast to the cases with \(m>2\), these two constructions are mutually consistent for \(m=2\). In that precise sense, the long-range Lee-Yang model is presented as the non-unitary analogue of the long-range Ising model [2603.27031].

## 1. Short-range Lee-Yang theory and its non-unitary CFT origin

The short-range Lee-Yang model is the simplest member of the non-unitary minimal-model series \(\mathcal{M}(2,2m+1)\). For \(m=2\), this is \(\mathcal{M}(2,5)\), whose spectrum contains only the identity and one nontrivial Virasoro primary \(\phi_{1,2}\) with scaling dimension
\[
\Delta_{1,2}=-\frac{4}{5}.
\]
The same model is also described in Landau-Ginzburg language by a scalar theory with imaginary cubic interaction \(i\varphi^3\). The action is antilinear/PT-symmetric, with the property \(S^*[-]=S[+]\), which is the mechanism proposed to keep the spectrum real despite the interaction being imaginary [2603.27031].

Within continuum field theory, the Lee-Yang edge singularity is also studied as the single real scalar cubic theory
\[
\mathcal L = (\partial \varphi)^2 + \lambda \varphi^3,
\]
with upper critical dimension \(d_c=6\). In this formulation, the exact two-dimensional fixed point is again \(\mathcal M(2,5)\), and the model is treated as a nonunitary critical theory whose cubic coupling is effectively imaginary at criticality [1612.08739].

These two descriptions provide the short-range starting point for the long-range problem. The long-range deformation does not replace the Lee-Yang model; rather, it asks whether a nonlocal theory with a tunable range parameter can reproduce the same infrared structure in a controlled perturbative regime.

## 2. Direct long-range deformation by a generalized free field

The first construction begins with a generalized free scalar field endowed with the nonlocal kinetic term
\[
\int d^d x_1\, d^d x_2\;\frac{\big(\phi(x_1)-\phi(x_2)\big)^2}{|x_1-x_2|^{d+s}},
\]
for which the scaling dimension is
\[
\Delta_\phi=\frac{d-s}{2}.
\]
In \(d=2\), this becomes \(\Delta_\phi=(2-s)/2\). The theory is nonlocal by construction and therefore does not have a local stress tensor or Virasoro symmetry. Its role is to introduce a continuous parameter \(s\) that lets one interpolate between mean-field and short-range behavior while keeping perturbation theory under better control than at the strongly coupled two-dimensional fixed point itself [2603.27031].

For the Lee-Yang case, the direct long-range action is
\[
S_{\rm LR}=S_{\rm GFF}[\phi]+\frac{i g_0}{3!}\int d^2x\,\phi^3.
\]
More generally, for \(\mathcal{M}(2,2m+1)\) one considers
\[
S_{\rm LR}^{(m)}=S_{\rm GFF}[\phi]+\frac{i g_0}{(2m-1)!}\int d^2x\,\phi^{2m-1}.
\]
Near marginality, where \((2m-1)\Delta_0=2-\epsilon\), the beta function has the form
\[
B(g)=-\epsilon g + B_3 g^3 + O(g^5),
\]
with only odd powers because of the \(Z_2\) symmetry. For the Lee-Yang case, the paper writes
\[
B(\lambda)= -\epsilon \lambda + \frac{272}{27}\lambda^3 + O(\lambda^5),
\]
and obtains a pair of imaginary fixed points. This is used to argue that the long-range Lee-Yang infrared theory is a real Euclidean QFT even though the coupling is imaginary [2603.27031].

A structurally important feature of this construction is the exact shadow relation
\[
\Delta_{\phi^2}\big|_{\ast}=2-\Delta_\phi,
\]
which is protected by the nonlocal Schwinger-Dyson equations rather than by local descendant structure. This relation becomes one of the key tests for comparing the direct long-range formulation with the alternative minimal-model-based construction.

## 3. Long-range deformation built from the minimal model

The second construction starts not from the generalized free field alone, but from the short-range Lee-Yang CFT itself coupled to a nonlocal generalized free field \(x\). For the Lee-Yang model this is written as
\[
S_{\rm LR}=S_{\mathcal{M}(2,5)}+S_{\rm GFF}[x]+g_0\int d^2x\,\phi_{1,2}\,x.
\]
For the general multicritical series, the analogous deformation is
\[
S_{\rm LRMM}^{(m)}=S_{\mathcal{M}(2,2m+1)}+S_{\rm GFF}[x]+g_0\int d^2x\,\phi_{1,2}\,x.
\]
Here \(\phi_{1,2}\) is the first nontrivial primary, identified as the primary with smallest absolute conformal dimension, and the generalized free field \(x\) is chosen so that the product \(\phi_{1,2}x\) has near-marginal dimension [2603.27031].

For the Lee-Yang case, the crossover point is
\[
s_\ast=\frac{14}{5}.
\]
If one integrates out \(x\), one obtains a nonlocal bilinear kernel of the form
\[
\sim -\int d^2x_1\,d^2x_2\;\frac{\phi_{1,2}(x_1)\phi_{1,2}(x_2)}{|x_1-x_2|^{2\Delta_x}},
\]
which is the analogue of the long-range kinetic term in the direct generalized-free-field description. In this formulation, \(\phi_{1,2}\) plays the role of the order parameter field, while \(x\) is its shadow partner. The paper stresses that this correspondence is only necessary, not fully proven nonperturbatively [2603.27031].

This second route is essential because it gives an independent candidate long-range realization of the Lee-Yang universality class. The long-range Lee-Yang model is therefore not a single Lagrangian ansatz but a consistency problem: the direct and minimal-model-based constructions should agree on fixed points, operator identifications, and stability.

## 4. Perturbative fixed points, operator dimensions, and consistency tests

In the minimal-model-based construction, the beta function again takes the form
\[
B(g)=-\epsilon g + B_3 g^3 + O(g^5),
\]
but now the coefficient \(B_3\) is extracted from a four-point integral in the unperturbed theory \(\text{GFF}\times \mathcal{M}(2,5)\). For the Lee-Yang case, the coefficient is evaluated numerically as
\[
B_3=(1.4019049631\pm 5\times10^{-10})>0.
\]
This yields a pair of real nontrivial fixed points,
\[
g_\ast^2=\frac{\epsilon}{B_3}.
\]
At the same fixed point, the anomalous dimension of \(\phi_{1,2}\) is fixed by the shadow relation
\[
\Delta_{\phi_{1,2}}\big|_\ast = 2-\Delta_x = \epsilon,
\]
and this is reported to be confirmed numerically to machine precision [2603.27031].

The stress tensor also acquires a perturbative anomalous dimension through multiplet recombination. The significance of this statement in the paper is that the resulting scaling behavior is consistent with stability for the Lee-Yang case. The special role of \(m=2\) is then formulated as a consistency statement: the direct long-range Landau-Ginzburg description and the long-range minimal-model deformation both produce compatible real infrared fixed points, the operator identifications line up, and the protected dimensions match [2603.27031].

From the broader Lee-Yang literature, this perturbative picture sits naturally beside functional-renormalization-group studies of the short-range cubic theory. Those studies emphasize that the Lee-Yang model has upper critical dimension \(6\), possesses two relevant exponents, and can be tracked continuously in dimension, although quantitative control becomes increasingly difficult as the dimension decreases [1612.08739]. This background explains why a long-range formulation with a tunable parameter \(s\) is technically attractive: it supplies a controlled deformation away from the strongly coupled two-dimensional limit.

## 5. Relation to the long-range Ising model and to Lee-Yang zeros

A central conclusion of the 2026 analysis is that the long-range Lee-Yang model is directly analogous to the long-range Ising model. The comparison is explicit: in both cases there is a long-range-to-short-range crossover controlled by \(s\), a shadow-partner structure between the order parameter and an auxiliary or nonlocal field, and a stable long-range fixed point whose perturbative data agree on both sides of the dual description [2603.27031].

A common source of ambiguity is that the phrase “Lee-Yang” is also used in the theory of partition-function zeros. In that separate setting, the Lee-Yang property of a spin model means that its partition function has purely imaginary zeros as a function of an external magnetic field. For isotropic vector ferromagnets on \(\mathds{Z}\), a generalized formulation expresses the partition function as
\[
Z_N(z)=Z_N(0)\prod_{j=1}^\infty (1+\gamma_{j,N}z^2),
\qquad \gamma_{j,N}>0,
\]
so that zeros occur when \(z^2=-1/\gamma_{j,N}\). That result concerns one-dimensional chains with nearest-neighbor ferromagnetic coupling and a strongly isotropic single-spin measure; it is not the same object as the long-range deformation of the non-unitary minimal model \(\mathcal M(2,5)\) [2603.18675].

The distinction is sharpened further by work on experimental detection of Lee-Yang zeros in long-range Ising baths. In that setting the bath Hamiltonian is an all-to-all ferromagnetic Ising model with
\[
J_{ij}=\frac{J}{N},
\]
and a probe spin is used to measure the analytically continued partition function through its time evolution. There, “long-range” refers to the interaction range in the Ising bath, and “Lee-Yang” refers to the zero structure of the partition function in complex magnetic field. The paper concludes that, even for long-range interacting baths, Lee-Yang zeros can be detected experimentally through probe-spin evolution without directly realizing complex magnetic fields [1810.01663]. This is conceptually adjacent, but it is a different usage from the long-range Lee-Yang model of non-unitary CFT and nonlocal field theory.

## 6. Generalizations beyond \(m=2\) and the status of the program

For the general family \(\mathcal{M}(2,2m+1)\), the direct long-range Landau-Ginzburg analysis continues to produce a pair of complex-conjugate fixed points at leading order, with \(B_3>0\) for all \(m\ge 2\). It also yields operator-dimension relations such as
\[
\Delta_{\phi^{2m-1}}\big|_\ast = 2+2\epsilon+O(\epsilon^2),
\qquad
\Delta_{\phi^{2m-2}}\big|_\ast = 2-\Delta_\phi.
\]
However, when the minimal-model-based construction is repeated for generic \(m\), the perturbative coefficient changes sign:
\[
B_3<0 \qquad \text{for } m>2.
\]
This sign change is the first major obstruction identified in the paper [2603.27031].

The paper then describes two concrete inconsistencies for \(m>2\). First, the fixed point coupling implied by
\[
g_\ast^2=\frac{\epsilon}{B_3}
\]
has the wrong sign in the induced nonlocal kernel after integrating out the auxiliary field \(x\), so the effective action is no longer of the correct long-range type and is not bounded below in the expected way. Second, the Virasoro stress-energy tensor becomes slightly relevant in the infrared,
\[
\Delta_T\big|_{g=g_\ast}=2+\gamma_T \epsilon+O(\epsilon^2),
\qquad \gamma_T<0 \quad (m>2),
\]
which makes the fixed point unstable. The paper therefore concludes that the \(m=2\) case appears to be the only member of the family for which the two proposed long-range descriptions are mutually consistent [2603.27031].

The resulting picture is sharply asymmetric. The long-range Lee-Yang model passes the consistency tests and behaves as a stable long-range realization of the Lee-Yang universality class. For higher multicritical Yang-Lee models, the perturbative evidence suggests that the conjectured correspondence breaks down. A plausible implication is that these higher cases require a more subtle interpretation or a different fixed-point structure.

Source: https://www.emergentmind.com/topics/long-range-lee-yang-model