Yang-Lee Model in Non-Unitary Field Theory

• The Yang-Lee model is a non-unitary quantum field theory that describes critical edge singularities in partition function zeros of Ising-type models.
• It employs integrable structures and minimal conformal field theory (M(2,5)) to characterize scaling behaviors and phase transitions in both quantum and statistical systems.
• Recent advances include experimental realizations and numerical validations using Bethe Ansatz, tensor networks, and finite-size scaling to probe its universal properties.

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 $h_{1,2}=-1/5$) (Cruz et al., 9 May 2025, Hollo et al., 2014, Bajnok et al., 2014).

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 (Zhang et al., 15 Oct 2025, Mussardo et al., 2017). 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 (Sanno et al., 2022, Shen et al., 2023): $$H_{\rm YL} = -J\sum_j \sigma^z_j \sigma^z_{j+1} - h_x\sum_j \sigma^x_j - i h_z \sum_j \sigma^z_j$$

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., $C_{\phi\phi\phi} \approx 1.9113\,i$ (Cruz et al., 9 May 2025)
• Spectrum of conformal dimensions organizes universal scaling and finite-size corrections in the Yang-Lee model and related chains (Bajnok et al., 2014)

Field-theoretic analysis by $6-\epsilon$ expansion of the $i\phi^3$ theory gives the anomalous dimensions over $2 \leq d < 6$ (Cruz et al., 9 May 2025). 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 $c=-57/7$, $\Delta_{1,2}=-2/7$, $-3/7$ (Lencsés et al., 2024).

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 (Garcia-Saez et al., 2015, Sedik et al., 2023). For antiferromagnets, the distribution is richer, with root curves tracing phase boundaries between paramagnetic and staggered-magnetized (complex-magnetization) phases (Abdelshafy et al., 18 Mar 2025).

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) (Dalmazi et al., 2010).

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 (Bajnok et al., 2014, Hollo et al., 2014). 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 (Hollo et al., 2014).

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 $T\to\infty$ (Mussardo et al., 2017).

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 (Zhang et al., 15 Oct 2025, Li, 2024). 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 (Xu et al., 2024).

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 (Shen et al., 2023, Zhang et al., 15 Oct 2025). In topological systems, non-unitary Yang-Lee anyons have been engineered in nanowire/Majorana platforms and their fusion, measurement, and braiding protocols analyzed (Sanno et al., 2022).

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 (Lencsés et al., 2024). 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 (Glumac et al., 2013).

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 (Cruz et al., 9 May 2025).

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 $\mathcal{M}(3,5)$ (Ardonne et al., 2010). 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 (Sanno et al., 2022).

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References:

• Explicit boundary form factors: (Hollo et al., 2014)
• Finite-volume spectra and integrability: (Bajnok et al., 2014)
• Yang-Lee zeros of the Yang-Lee model: (Mussardo et al., 2017)
• Ginzburg-Landau multicritical Yang-Lee: (Lencsés et al., 2024)
• Yang-Lee criticality in dimensions $d=1$–$4$: (Cruz et al., 9 May 2025)
• Quantum criticality in non-Hermitian systems: (Zhang et al., 15 Oct 2025, Li, 2024, Xu et al., 2024)
• Tensor-network and cluster expansion methods: (Garcia-Saez et al., 2015, Abdelshafy et al., 18 Mar 2025, Sedik et al., 2023)
• Anyonic chain realizations: (Ardonne et al., 2010, Sanno et al., 2022)
• Experimental proposals: (Shen et al., 2023)

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