---
title: Yang-Lee Model in Non-Unitary Field Theory
url: https://www.emergentmind.com/topics/yang-lee-model
type: topic
---

# Yang-Lee Model in Non-Unitary Field Theory

The Yang-Lee model, also known as the scaling Lee-Yang model or Yang-Lee edge singularity field theory, occupies a central position among non-unitary integrable quantum field theories (QFTs) and statistical models. It controls the universal critical singularities that arise at the edge of the partition function zero locus in Ising-type and Potts-type models subjected to imaginary external fields. Its critical point is described by the non-unitary minimal conformal field theory (CFT) $\mathcal{M}(2,5)$, with generalizations to multicritical points realized by other non-unitary minimal models. The Yang-Lee model also appears as the effective field theory for quantum and classical non-Hermitian systems with parity-time ($\mathcal{PT}$) symmetry, in particular at $\mathcal{PT}$-breaking transitions. Modern developments encompass rigorous integrable-structure analysis, exact finite-volume spectra, form factors, tensor-network and linked-cluster methods for partition function zeros, quantum entanglement transitions, and direct experimental implementation in quantum simulators and engineered Majorana platforms.

## 1. Definition: Field Theory, Lattice Realizations, and Partition Function Zeros

The Yang-Lee model is most fundamentally realized as the non-unitary Euclidean field theory
\[
S[\phi] = \int d^d x\, \left[ \frac{1}{2} (\partial_\mu \phi)^2 + i\, (h-h_0)\, \phi + i\, g\, \phi^3 \right]
\]
with $g \in i\mathbb{R}$; this is the $i\phi^3$ field theory analyzed by Fisher. In $d=2$, at $h=h_0$ and $g=g_c$, the theory is integrable and coincides in the ultraviolet with the non-unitary minimal model $\mathcal{M}(2,5)$ (central charge $c=-22/5$, relevant field $\phi \equiv \Phi_{1,2}$, scaling dimension $h_{1,2}=-1/5$) [2505.06369, 1405.3820, 1412.8494].

On the lattice, the Yang-Lee universality class is realized as the edge singularity of the partition function zeros for the Ising model in a complex (imaginary) magnetic field:
\[
Z(h) = \sum_{\{s_i\}} \exp\left[-\beta H_0(\{s_i\}) + \beta h \sum_i s_i\right]
\]
where $H_0$ is an Ising-type Hamiltonian and $h\in\mathbb{C}$. In the thermodynamic limit, the Lee–Yang zeros form loci in the complex-$h$ plane; their endpoints (Yang-Lee edges) are critical branch points with nontrivial exponents governed by the Yang-Lee model [2510.13581, 1708.06444]. The edge singularity is analytically continued “through” the phase transition, giving a non-unitary fixed point without physical symmetry-breaking.

More generally, the model governs quantum criticality in non-Hermitian, $\mathcal{PT}$-symmetric quantum spin systems, such as the transverse-field Ising chain with an imaginary longitudinal field [2208.01803, 2302.06662]:
\[
H_{\rm YL} = -J\sum_j \sigma^z_j \sigma^z_{j+1} - h_x\sum_j \sigma^x_j - i h_z \sum_j \sigma^z_j
\]

## 2. Conformal Field Theory Data, Exponents, Multicritical Generalizations

The Yang-Lee critical point in $d=2$ is described exactly by the minimal model $\mathcal{M}(2,5)$, characterized by:
- Central charge $c=-22/5$
- Two primary fields: identity ($h=0$) and $\phi \equiv \Phi_{1,2}$ ($h=-1/5$)
- Fusion algebra: $\phi \times \phi = 1 + \phi$
- OPE structure constants, e.g., $C_{\phi\phi\phi} \approx 1.9113\,i$ [2505.06369]
- Spectrum of conformal dimensions organizes universal scaling and finite-size corrections in the Yang-Lee model and related chains [1412.8494]

Field-theoretic analysis by $6-\epsilon$ expansion of the $i\phi^3$ theory gives the anomalous dimensions over $2 \leq d < 6$ [2505.06369]. In $d=3$, fits yield $\Delta_\phi \approx 0.214$, $\Delta_{\phi^3}\approx 4.61$, matching “fuzzy sphere” and Platonic solid numerics.

Multicritical Yang-Lee singularities generalize to potentials of the form $\varphi^2 (i\varphi)^n$; at $n=2$, the tricritical point flows to $\mathcal{M}(2,7)$ with $c=-57/7$, $\Delta_{1,2}=-2/7$, $-3/7$ [2404.06100].

## 3. Partition Function Zeros, Yang-Lee Edge Singularities, and Critical Exponents

In ferromagnetic Ising models, the Lee–Yang theorem asserts that all zeros of $Z(z)$, $z = e^{2\beta h}$, lie on the unit circle $|z|=1$. The edge singularity $h_c$ is the point on the imaginary-$h$ axis where zeros pinch the real axis as temperature is lowered [1508.07751, 2309.14562]. For antiferromagnets, the distribution is richer, with root curves tracing phase boundaries between paramagnetic and staggered-magnetized (complex-magnetization) phases [2503.14602].

Critical exponents at the Yang-Lee edge in $d=2$ are:
\[
\sigma = -1/6 \quad \text{(density of zeros)};\quad \Delta_\phi = -2/5
\]
Second-order phase transitions align with zeros pinching the real axis; exponents can be extracted by finite-size scaling of the zeros’ accumulation rate, with $\Im h \propto |\Re h-h_c|^\sigma$.

Higher-order and multicritical exponents arise at “tricritical” points where transfer matrix eigenvalues become triply degenerate ($\sigma=-2/3$ in $d=1$ fine-tuned models) [1012.4454].

## 4. Exact Solutions: Integrability, Bethe Ansatz, Boundary and Finite-Volume Methods

The Yang-Lee model is integrable in $d=2$ both in the bulk and on the boundary. Its spectrum consists of a single self-conjugate massive particle with the exact $S$-matrix:
\[
S_{YL}(\theta) = \frac{\sinh \theta + i \sin(2\pi/3)}{\sinh \theta - i \sin(2\pi/3)}
\]
with a pole at $\theta=2\pi i/3$ indicating a bound state [1412.8494, 1405.3820]. The thermodynamic Bethe ansatz yields the grand-canonical partition function, compressibility, and thus the Yang-Lee zeros.

Boundary integrable variants admit identity and $\phi$-type boundaries with explicit reflection amplitudes, classified by associated conformal boundary conditions. All exact boundary form factors for local operators can be constructed recursively and, for the stress-tensor, expressed in determinant form. For the primary boundary field $\phi$, additional polynomials enter [1405.3820].

The spectral density of the zeros is computable via TBA-based cluster expansions. In contrast to the free-fermion case where zeros remain at $|z|\approx1$ for $T>0$, the radius for the Yang-Lee model smoothly shrinks to zero as $T\to\infty$ [1708.06444].

## 5. Quantum Phase Transitions, Entanglement, and Non-Hermitian/Experimental Realizations

The Yang-Lee universality class controls non-Hermitian quantum phase transitions, notably at $\mathcal{PT}$-symmetry-breaking exceptional points in quantum chains and models such as the non-Hermitian PXP model or the quantum Ising chain with an imaginary field [2510.13581, 2407.00311]. At the critical point, the system exhibits Ising universality class scaling in the real regime and Yang-Lee CFT exponents at the edge. The “edge” can be sharply detected via biorthogonal Loschmidt echoes, ground-state entanglement transitions, and dynamical quantum phase transitions [2412.07800].

Yang-Lee criticality has been proposed and soon realized experimentally in Rydberg atomic arrays, both via dynamical protocols and direct engineering of $\mathcal{PT}$-symmetric or open-system Hamiltonians [2302.06662, 2510.13581]. In topological systems, non-unitary Yang-Lee anyons have been engineered in nanowire/Majorana platforms and their fusion, measurement, and braiding protocols analyzed [2208.01803].

## 6. Generalizations: Multicriticality, Potts Models, and Higher Dimensions

Fisher’s argument extends the Yang-Lee singularity to a broader class of multicritical points with non-Hermitian Ginzburg–Landau potentials $V(\varphi) = \varphi^2 (i\varphi)^n$, matching the minimal models $\mathcal{M}(2,2n+3)$ at relevant critical couplings [2404.06100]. For $n=2$, the “tricritical Yang-Lee” point corresponds to $\mathcal{M}(2,7)$ and arises as a boundary of the Yang-Lee line in coupling space.

In Potts models (e.g., $q=3$), the phase diagram exhibits KMS lines, tricritical points, and explicitly calculable root loci. The density of zeros, scaling, and phase boundaries are obtainable via saddle-point and cluster expansion methods [1302.7197].

In $d>2$, the upper critical dimension of the Yang-Lee universality class is $d=6$. The $6-\epsilon$ expansion of $i\phi^3$-theory gives scaling dimensions and OPE coefficients, which can be numerically validated by diagonalization on Platonic solids, fuzzy spheres, and related regularizations. In $d=3$, numerical spectra and correlation data agree precisely with high-temperature expansions and Padé-resummed perturbative results; in $d=4$, similar methods using the 24-cell triangulation extend this quantitative test [2505.06369].

## 7. Algebraic Structure, Anyonic Realizations, and Topological Phases

Chain Hamiltonians of “Yang-Lee anyons”—the Galois conjugates of Fibonacci anyons—define non-unitary versions of the golden chain or Levin–Wen models. Their fusion algebra, quantum dimensions, and explicit F- and R-symbols are fixed by the Yang-Lee CFT, and they manifest gapless non-unitary critical points described by $\mathcal{M}(2,5)$ or $\mathcal{M}(3,5)$ [1012.1080]. The criticality is topologically protected by nontrivial symmetries, and the platform supports non-unitary braiding statistics, opening a path to measurement-based quantum gates with non-Hermitian dynamics [2208.01803].

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**References:**
- Explicit boundary form factors: [1405.3820]
- Finite-volume spectra and integrability: [1412.8494]
- Yang-Lee zeros of the Yang-Lee model: [1708.06444]
- Ginzburg-Landau multicritical Yang-Lee: [2404.06100]
- Yang-Lee criticality in dimensions $d=1$–$4$: [2505.06369]
- Quantum criticality in non-Hermitian systems: [2510.13581, 2407.00311, 2412.07800]
- Tensor-network and cluster expansion methods: [1508.07751, 2503.14602, 2309.14562]
- Anyonic chain realizations: [1012.1080, 2208.01803]
- Experimental proposals: [2302.06662]

The Yang-Lee model thus embodies a universal and exactly tractable framework for non-unitary critical phenomena, spanning statistical, quantum, topological, and experimental frontiers.

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