Long-Range Lee-Yang Model

• The long-range Lee-Yang model is a non-unitary minimal model (m=2) variant characterized by long-range deformations and an imaginary cubic interaction.
• It is studied via two constructions: one using a generalized free scalar field and another coupling the Lee-Yang CFT with a nonlocal field, both yielding matching infrared fixed points.
• The analysis confirms that only the m=2 case produces consistent operator dimensions and stable fixed points, unlike higher multicritical Yang-Lee models.

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 (Eustachon, 27 Mar 2026).

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 $\mathcal{M}(2,2m+1)$0. The action is antilinear/PT-symmetric, with the property $\mathcal{M}(2,2m+1)$1, which is the mechanism proposed to keep the spectrum real despite the interaction being imaginary (Eustachon, 27 Mar 2026).

Within continuum field theory, the Lee-Yang edge singularity is also studied as the single real scalar cubic theory

$\mathcal{M}(2,2m+1)$2

with upper critical dimension $\mathcal{M}(2,2m+1)$3. In this formulation, the exact two-dimensional fixed point is again $\mathcal{M}(2,2m+1)$4, and the model is treated as a nonunitary critical theory whose cubic coupling is effectively imaginary at criticality (Zambelli et al., 2016).

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

$\mathcal{M}(2,2m+1)$5

for which the scaling dimension is

$\mathcal{M}(2,2m+1)$6

In $\mathcal{M}(2,2m+1)$7, this becomes $\mathcal{M}(2,2m+1)$8. 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 $\mathcal{M}(2,2m+1)$9 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 (Eustachon, 27 Mar 2026).

For the Lee-Yang case, the direct long-range action is

$\mathcal{M}(2,5)$0

More generally, for $\mathcal{M}(2,5)$1 one considers

$\mathcal{M}(2,5)$2

Near marginality, where $\mathcal{M}(2,5)$3, the beta function has the form

$\mathcal{M}(2,5)$4

with only odd powers because of the $\mathcal{M}(2,5)$5 symmetry. For the Lee-Yang case, the paper writes

$\mathcal{M}(2,5)$6

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 (Eustachon, 27 Mar 2026).

A structurally important feature of this construction is the exact shadow relation

$\mathcal{M}(2,5)$7

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 $\mathcal{M}(2,5)$8. For the Lee-Yang model this is written as

$\mathcal{M}(2,5)$9

For the general multicritical series, the analogous deformation is

$m>2$0

Here $m>2$1 is the first nontrivial primary, identified as the primary with smallest absolute conformal dimension, and the generalized free field $m>2$2 is chosen so that the product $m>2$3 has near-marginal dimension (Eustachon, 27 Mar 2026).

For the Lee-Yang case, the crossover point is

$m>2$4

If one integrates out $m>2$5, one obtains a nonlocal bilinear kernel of the form

$m>2$6

which is the analogue of the long-range kinetic term in the direct generalized-free-field description. In this formulation, $m>2$7 plays the role of the order parameter field, while $m>2$8 is its shadow partner. The paper stresses that this correspondence is only necessary, not fully proven nonperturbatively (Eustachon, 27 Mar 2026).

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

$m>2$9

but now the coefficient $m=2$0 is extracted from a four-point integral in the unperturbed theory $m=2$1. For the Lee-Yang case, the coefficient is evaluated numerically as

$m=2$2

This yields a pair of real nontrivial fixed points,

$m=2$3

At the same fixed point, the anomalous dimension of $m=2$4 is fixed by the shadow relation

$m=2$5

and this is reported to be confirmed numerically to machine precision (Eustachon, 27 Mar 2026).

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$6 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 (Eustachon, 27 Mar 2026).

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 $m=2$7, possesses two relevant exponents, and can be tracked continuously in dimension, although quantitative control becomes increasingly difficult as the dimension decreases (Zambelli et al., 2016). This background explains why a long-range formulation with a tunable parameter $m=2$8 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 $m=2$9, 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 (Eustachon, 27 Mar 2026).

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 $\mathcal{M}(2,2m+1)$0, a generalized formulation expresses the partition function as

$\mathcal{M}(2,2m+1)$1

so that zeros occur when $\mathcal{M}(2,2m+1)$2. 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}(2,2m+1)$3 (Kozitsky, 19 Mar 2026).

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

$\mathcal{M}(2,2m+1)$4

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 (Kuzmak et al., 2018). 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 $\mathcal{M}(2,2m+1)$5 and the status of the program

For the general family $\mathcal{M}(2,2m+1)$6, the direct long-range Landau-Ginzburg analysis continues to produce a pair of complex-conjugate fixed points at leading order, with $\mathcal{M}(2,2m+1)$7 for all $\mathcal{M}(2,2m+1)$8. It also yields operator-dimension relations such as

$\mathcal{M}(2,2m+1)$9

However, when the minimal-model-based construction is repeated for generic $m=2$0, the perturbative coefficient changes sign: $m=2$1 This sign change is the first major obstruction identified in the paper (Eustachon, 27 Mar 2026).

The paper then describes two concrete inconsistencies for $m=2$2. First, the fixed point coupling implied by

$m=2$3

has the wrong sign in the induced nonlocal kernel after integrating out the auxiliary field $m=2$4, 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,

$m=2$5

which makes the fixed point unstable. The paper therefore concludes that the $m=2$6 case appears to be the only member of the family for which the two proposed long-range descriptions are mutually consistent (Eustachon, 27 Mar 2026).

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.