Lee–Yang Polynomial Essentials

• Lee–Yang polynomials are multivariate polynomials that exhibit no zeros in both the open unit polydisc and its inverse, ensuring stability with zeros confined to the boundary.
• They connect complex analysis, stability theory, and statistical mechanics by enforcing zero-location constraints, as exemplified in the Lee–Yang circle theorem for Ising models.
• Recent extensions of the theory include applications to Fourier quasicrystals, quantum tensor networks, and edge-coloured graph counting, highlighting their broad analytic impact.

A Lee–Yang polynomial is, in one standard multivariate sense, a polynomial $p(z_{1},\ldots,z_{n})$ that has no zeros in the open unit polydisc $\mathbb{D}^{n}$ and no zeros in the inverse polydisc $(\mathbb{C}\setminus\overline{\mathbb{D}})^{n}$ (Alon et al., 2023). In statistical mechanics, the same expression is also used for partition functions or moment generating functions whose zeros lie on distinguished loci, classically the unit circle in a fugacity variable or the imaginary axis after a change of variables (Newman et al., 2017). The term is therefore used in several related senses, but the common content is a strong zero-location constraint that connects complex analysis, stability theory, exponential polynomials, Fourier quasicrystals, orthogonal polynomials, and quantum or geometric extensions.

1. Core definitions and classical formulations

Let $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$. A multivariate polynomial $p\in\mathbb{C}[z_1,\ldots,z_n]$ is called Schur stable if it has no zeros in $\mathbb{D}^n$. Following Ruelle, a Lee–Yang polynomial is one that has no zeros in both $\mathbb{D}^n$ and $(\mathbb{C}\setminus\overline{\mathbb{D}})^n$. Equivalently, if

$$p^{\dagger}(z_1,\ldots,z_n):=\prod_{j=1}^{n} z_j^{\deg_j(p)}\,p(1/z_1,\ldots,1/z_n),$$

then $p$ is Lee–Yang exactly when both $\mathbb{D}^{n}$0 and $\mathbb{D}^{n}$1 are Schur stable (Alon et al., 2023).

This zero-free condition is stronger than Schur stability alone. It forces zeros away from complementary radial regions, so that any zero seen relative to the unit torus must lie on the boundary $\mathbb{D}^{n}$2. In the terminology of stability theory used in the cited work, Schur stability means nonvanishing on $\mathbb{D}^{n}$3, whereas Lee–Yang stability adds nonvanishing on the inverse polydisc. The same source emphasizes that this notion is distinct from “real-stable” or upper-half-plane stability, and that no positivity or reality condition on the coefficients is required in its definition (Alon et al., 2023).

In the classical statistical-mechanical formulation, the finite-volume partition function of a ferromagnetic Ising model is a polynomial in a fugacity variable $\mathbb{D}^{n}$4 or $\mathbb{D}^{n}$5, depending on normalization. The Lee–Yang circle theorem states that all zeros lie on the unit circle $\mathbb{D}^{n}$6. In the field variable $\mathbb{D}^{n}$7, this is equivalent to zeros on the imaginary axis, since $\mathbb{D}^{n}$8 if and only if $\mathbb{D}^{n}$9 (Newman et al., 2017).

2. Geometric structure, amoebas, and torus restrictions

A central geometric tool is the amoeba

$(\mathbb{C}\setminus\overline{\mathbb{D}})^{n}$0

Classical results of Gelfand–Kapranov–Zelevinsky imply that $(\mathbb{C}\setminus\overline{\mathbb{D}})^{n}$1 is a union of open convex regions. Because $(\mathbb{C}\setminus\overline{\mathbb{D}})^{n}$2 sends $(\mathbb{C}\setminus\overline{\mathbb{D}})^{n}$3 to $(\mathbb{C}\setminus\overline{\mathbb{D}})^{n}$4 and $(\mathbb{C}\setminus\overline{\mathbb{D}})^{n}$5 to $(\mathbb{C}\setminus\overline{\mathbb{D}})^{n}$6, one has:

• $(\mathbb{C}\setminus\overline{\mathbb{D}})^{n}$7 has no zeros in $(\mathbb{C}\setminus\overline{\mathbb{D}})^{n}$8 if and only if $(\mathbb{C}\setminus\overline{\mathbb{D}})^{n}$9,
• $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$0 has no zeros in $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$1 if and only if $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$2.

If $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$3, these conditions are equivalent to the Lee–Yang property: $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$4 This gives a geometric characterization of Lee–Yang nonvanishing in terms of full cones avoided by the amoeba (Alon et al., 2023).

The same framework naturally leads to torus restrictions. A positive line in the $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$5-torus is a map

$\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$6

where $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$7 has $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$8-linearly independent entries; that rational independence makes the line dense in the torus. Restricting a Lee–Yang polynomial to such a dense torus line produces a one-variable exponential polynomial with strong zero-location properties, and the geometry of the amoeba is precisely what links the multivariate zero-free regions to real-rootedness of that restriction (Alon et al., 2023).

3. Real-rooted exponential polynomials and Fourier quasicrystals

A principal theorem states that every real-rooted exponential polynomial

$\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$9

can be written as the restriction of a Lee–Yang polynomial to a positive line in a torus, up to a nonvanishing exponential factor. More precisely, if $p\in\mathbb{C}[z_1,\ldots,z_n]$0 and

$p\in\mathbb{C}[z_1,\ldots,z_n]$1

then there exist a Lee–Yang polynomial $p\in\mathbb{C}[z_1,\ldots,z_n]$2 and a vector $p\in\mathbb{C}[z_1,\ldots,z_n]$3 with $p\in\mathbb{C}[z_1,\ldots,z_n]$4-linearly independent entries such that

$p\in\mathbb{C}[z_1,\ldots,z_n]$5

The integer $p\in\mathbb{C}[z_1,\ldots,z_n]$6 is optimal in the $p\in\mathbb{C}[z_1,\ldots,z_n]$7-linear dimension sense (Alon et al., 2023).

The construction is explicit. Real-rootedness and a classical theorem of Pólya imply that the frequency differences $p\in\mathbb{C}[z_1,\ldots,z_n]$8 are purely imaginary, $p\in\mathbb{C}[z_1,\ldots,z_n]$9 with $\mathbb{D}^n$0. One then compresses the frequency vector $\mathbb{D}^n$1 into an integer matrix $\mathbb{D}^n$2 and a positive rationally independent vector $\mathbb{D}^n$3, writes $\mathbb{D}^n$4, and defines

$\mathbb{D}^n$5

A further monomial change of variables upgrades this polynomial to a Lee–Yang polynomial by forcing its amoeba to avoid $\mathbb{D}^n$6 (Alon et al., 2023).

An illustrative example is

$\mathbb{D}^n$7

For this $\mathbb{D}^n$8, the construction yields

$\mathbb{D}^n$9

and $\mathbb{D}^n$0 is Lee–Yang. Along the positive line

$\mathbb{D}^n$1

the restriction of $\mathbb{D}^n$2 recovers $\mathbb{D}^n$3 up to the factor $\mathbb{D}^n$4 (Alon et al., 2023).

This theorem completes the bridge to one-dimensional Fourier quasicrystals. Kurasov–Sarnak showed that if $\mathbb{D}^n$5 is Lee–Yang and $\mathbb{D}^n$6, then

$\mathbb{D}^n$7

is an $\mathbb{D}^n$8-valued Fourier quasicrystal, where $\mathbb{D}^n$9. Together with Olevskii–Ulanovskii, the restriction theorem implies the converse: a measure $(\mathbb{C}\setminus\overline{\mathbb{D}})^n$0 on $(\mathbb{C}\setminus\overline{\mathbb{D}})^n$1 is an $(\mathbb{C}\setminus\overline{\mathbb{D}})^n$2-valued Fourier quasicrystal if and only if $(\mathbb{C}\setminus\overline{\mathbb{D}})^n$3 for some Lee–Yang polynomial $(\mathbb{C}\setminus\overline{\mathbb{D}})^n$4 and some $(\mathbb{C}\setminus\overline{\mathbb{D}})^n$5 (Alon et al., 2023). Subsequent work proves that this construction generically yields non-periodic $(\mathbb{C}\setminus\overline{\mathbb{D}})^n$6-FQs with unit coefficients and uniformly discrete support, and that every $(\mathbb{C}\setminus\overline{\mathbb{D}})^n$7-FQ has a limiting gap distribution, with Poisson and CUE limits appearing in natural families (Alon et al., 2023).

4. Partition functions, moment generating functions, and zero sets in statistical mechanics

In statistical mechanics, Lee–Yang polynomials arise most directly as finite-volume partition functions in a complex external field. For ferromagnetic Ising models, zeros lie on the unit circle in the fugacity variable; for MGFs $(\mathbb{C}\setminus\overline{\mathbb{D}})^n$8, the corresponding Lee–Yang property is “pure imaginary zeros,” meaning $(\mathbb{C}\setminus\overline{\mathbb{D}})^n$9 whenever $$p^{\dagger}(z_1,\ldots,z_n):=\prod_{j=1}^{n} z_j^{\deg_j(p)}\,p(1/z_1,\ldots,1/z_n),$$0. The class $$p^{\dagger}(z_1,\ldots,z_n):=\prod_{j=1}^{n} z_j^{\deg_j(p)}\,p(1/z_1,\ldots,1/z_n),$$1 used in this setting requires symmetry $$p^{\dagger}(z_1,\ldots,z_n):=\prod_{j=1}^{n} z_j^{\deg_j(p)}\,p(1/z_1,\ldots,1/z_n),$$2, a sub-Gaussian tail condition $$p^{\dagger}(z_1,\ldots,z_n):=\prod_{j=1}^{n} z_j^{\deg_j(p)}\,p(1/z_1,\ldots,1/z_n),$$3 for some $$p^{\dagger}(z_1,\ldots,z_n):=\prod_{j=1}^{n} z_j^{\deg_j(p)}\,p(1/z_1,\ldots,1/z_n),$$4, and pure imaginary zeros. When $$p^{\dagger}(z_1,\ldots,z_n):=\prod_{j=1}^{n} z_j^{\deg_j(p)}\,p(1/z_1,\ldots,1/z_n),$$5, the MGF has the canonical product expansion

$$p^{\dagger}(z_1,\ldots,z_n):=\prod_{j=1}^{n} z_j^{\deg_j(p)}\,p(1/z_1,\ldots,1/z_n),$$6

This framework captures the Lee–Yang property for Ising-type observables and extends to ferromagnetic XY and Villain models; the same paper also shows that the property fails for complex Gaussian multiplicative chaos in the parameter range $$p^{\dagger}(z_1,\ldots,z_n):=\prod_{j=1}^{n} z_j^{\deg_j(p)}\,p(1/z_1,\ldots,1/z_n),$$7, where tail behavior obstructs pure imaginary zeros (Newman et al., 2017).

For the Curie–Weiss ferromagnet, the finite-$$p^{\dagger}(z_1,\ldots,z_n):=\prod_{j=1}^{n} z_j^{\deg_j(p)}\,p(1/z_1,\ldots,1/z_n),$$8 partition function is explicitly identified with a unitary Hermite polynomial. Writing $$p^{\dagger}(z_1,\ldots,z_n):=\prod_{j=1}^{n} z_j^{\deg_j(p)}\,p(1/z_1,\ldots,1/z_n),$$9, one has

$p$0

where

$p$1

All zeros of $p$2 lie on the unit circle for any $p$3 and $p$4, and the empirical zero distribution of $p$5 converges to the free unitary normal distribution (Kabluchko, 2022).

The antiferromagnetic setting is markedly different. For nearest-neighbor and mean-field Ising antiferromagnets, the logarithm of the Yang–Lee zeros has a high-temperature expansion in half odd integer powers of the inverse temperature $p$6, with leading term $p$7. In the mean-field antiferromagnetic case, the thermodynamic-limit zeros lie on noncircular root curves rather than a Lee–Yang circle, and these curves separate regions of zero and non-zero complex staggered magnetization (Sedik et al., 2023).

5. Terminological variants and distinct usages

The literature does not use the expression “Lee–Yang polynomial” uniformly. In the theory of planar orthogonal polynomials introduced by S.-Y. Lee and M. Yang, the term refers to the monic planar orthogonal polynomial $p$8 associated with the modified Gaussian measure

$p$9

Lee and Yang showed that $\mathbb{D}^{n}$00 is a type II multiple orthogonal polynomial on a contour. When the exponents $\mathbb{D}^{n}$01 are positive integers, the same polynomials are also type I multiple orthogonal polynomials, and several equivalent Riemann–Hilbert formulations follow from the fundamental identity of Lee and Yang. This usage is explicitly distinguished from the polynomials of the Lee–Yang circle theorem and from stable polynomials in the sense of Borcea–Brändén (Berezin et al., 2022).

Another analogical use occurs in knot-theoretic and topological-field-theoretic contexts, where “Lee–Yang type” means that zeros lie on the unit circle in a complex variable $\mathbb{D}^{n}$02. In the cited SU(2) Chern–Simons examples, the normalized Wilson loop expectation identified with the Jones polynomial has zeros at $\mathbb{D}^{n}$03 for a single spin-$\mathbb{D}^{n}$04 loop and at $\mathbb{D}^{n}$05 for the Hopf link, all on $\mathbb{D}^{n}$06 (Zhou et al., 2019).

These variants do not collapse to a single universal definition. A persistent source of confusion is therefore terminological rather than mathematical: in some papers the phrase denotes a zero-free multivariate polynomial on the polydisc and inverse polydisc, in others a one-variable partition-function polynomial with unit-circle zeros, and in others a specific family of orthogonal polynomials attached to the work of S.-Y. Lee and M. Yang (Alon et al., 2023).

6. Geometric, combinatorial, and quantum extensions

Recent work extends Lee–Yang structures well beyond their original statistical-mechanical setting. For $\mathbb{D}^{n}$07-periodic $\mathbb{D}^{n}$08-hypersurfaces $\mathbb{D}^{n}$09, a Fourier criterion formulated through the directional measure

$\mathbb{D}^{n}$10

and a cone-supported Fourier transform $\mathbb{D}^{n}$11 implies that $\mathbb{D}^{n}$12 is algebraic of torus type: $\mathbb{D}^{n}$13 for an essentially Lee–Yang polynomial $\mathbb{D}^{n}$14, meaning that after a suitable monomial change of variables $\mathbb{D}^{n}$15 can be taken Lee–Yang. The cone-support hypothesis is described using Meyer’s terminology as a “lighthouse” (Alon et al., 21 Jul 2025).

A tensor-theoretic generalization identifies a complex tensor with $\mathbb{D}^{n}$16 binary indices with a multilinear polynomial in $\mathbb{D}^{n}$17 variables and calls it a Lee–Yang tensor with radius $\mathbb{D}^{n}$18 if the polynomial is nonzero whenever all variables lie in the open disk of radius $\mathbb{D}^{n}$19. The class is closed under tensor contraction and certain quantum operations. For $\mathbb{D}^{n}$20, the cited work proves that the corresponding quantum states can be prepared by quasipolynomial-sized circuits and that every Hermitian operator with Lee–Yang radius $\mathbb{D}^{n}$21 has a unique principal eigenvector. The same paper studies a two-local Hamiltonian favoring the deformed EPR state $\mathbb{D}^{n}$22, and numerically finds ground-state radius at least $\mathbb{D}^{n}$23 and spectral gap at least $\mathbb{D}^{n}$24 on all graphs considered (Wong et al., 3 Feb 2026).

In one-dimensional isotropic vector ferromagnets and lattice fields, the generalized Lee–Yang property takes the form

$\mathbb{D}^{n}$25

so zeros are confined to $\mathbb{D}^{n}$26. The paper establishes this for all even $\mathbb{D}^{n}$27 on $\mathbb{D}^{n}$28, extending the previously known $\mathbb{D}^{n}$29 case to isotropic spin and field models living on the one-dimensional lattice (Kozitsky, 19 Mar 2026).

A different extension appears in edge-coloured graph counting. The polynomial

$\mathbb{D}^{n}$30

specializes to the partition function of the ferromagnetic Ising model on a random regular graph. Its zeros accumulate, as $\mathbb{D}^{n}$31, along semialgebraic anti-Stokes curves arising from a saddle-point analysis of an exponential integral. The paper describes this zero accumulation as a Lee–Yang phenomenon in analogy with the classical theorem (Wiesmann, 5 Jan 2026).

Taken together, these developments show that Lee–Yang polynomials now function as a broad analytic paradigm. The strongest common theme is the control of zeros by a rigid geometric constraint—unit circles, slit planes, complementary polydiscs, torus hypersurfaces, or cone-supported Fourier data—while the concrete meaning of the term continues to depend on the surrounding framework.