Lee-Yang Property in Statistical Mechanics

• Lee–Yang property is a concept where the zeros of a partition function are confined to specific loci in the complex plane, dictating analyticity and phase transitions.
• It is applied in both equilibrium and dynamical settings, linking the geometry of complex zeros with critical scaling and universal transition exponents.
• Experimental techniques and advanced numerical methods validate the theory, mapping zeros via probe spin coherence and simulation in various statistical and quantum models.

The Lee–Yang property is the statement that the zeros of a partition function, or of an analogous generating function, occupy a sharply constrained locus in a complexified control parameter and thereby control analyticity, criticality, and phase structure. In its original form, for ferromagnetic Ising systems the zeros in the fugacity variable lie on the unit circle, equivalently on the imaginary axis of the magnetic field; in a broader usage, the term also covers multivariate, dynamical, and operator-theoretic settings in which the geometry, accumulation, and scaling of complex zeros encode equilibrium or trajectory-space phase transitions (Peng et al., 2014, Brandner et al., 2016, Fröhlich et al., 2012).

1. Classical formulation and circle theorems

For a finite equilibrium system with Hamiltonian $H(h)$ in external field $h$, the partition function is

$Z(\beta,h)=\mathrm{Tr}\,e^{-\beta H(h)}.$

In the ferromagnetic Ising case one rewrites $Z$ as a polynomial in a fugacity-like variable such as $z=e^{-\beta h}$ or $z=e^{-2\beta h}$. The Lee–Yang zeros are the complex roots $z_n$ of that polynomial. Because the Boltzmann weights are positive for real $h$, zeros do not occur at physical real fields for finite volume; instead, the free energy becomes singular only when, in the thermodynamic limit, the zeros condense onto sets that touch the real axis (Peng et al., 2014, Deger et al., 2019).

The classical Lee–Yang circle theorem states that for ferromagnetic Ising models with real, positive couplings, all zeros lie on the unit circle in the complex fugacity plane. In equivalent field language, the zeros lie on the imaginary axis of $h$. The theorem was later generalized to higher-spin ferromagnetic Ising systems and to other ferromagnetic models, while Fisher zeros extend the same analytic program to complex temperature (Peng et al., 2014).

A useful factorization is

$$Z(\beta,h)=p_0\,e^{\beta N h/2}\prod_{n=1}^{N}(z-z_n),$$

so the reduced free energy per site,

$h$0

is analytic away from the zero set. The Lee–Yang property is therefore not merely a statement about root location; it is a precise encoding of the analyticity domain of thermodynamic potentials and of the mechanism by which phase transitions emerge in the infinite-volume limit (Peng et al., 2014, Fröhlich et al., 2012).

2. Analyticity, edges, and critical scaling

The most immediate consequence of the Lee–Yang property is analyticity of thermodynamic observables for complex fields outside the zero locus. For lattice systems satisfying a Lee–Yang theorem, connected correlations are analytic functions of the external field for $h$1 or $h$2. In the Ising-type setting this analyticity extends to thermodynamic limits of connected correlations, and it underlies results on boundary-condition independence, positivity of the mass gap, and absence of phase coexistence away from the Lee–Yang boundary (Fröhlich et al., 2012).

Above the critical temperature, the Lee–Yang zeros leave a gap around the physical axis; the endpoints of that gap are the Yang–Lee edge singularities. At or below the critical temperature, the gap closes and the accumulation set reaches the real axis, producing a thermodynamic singularity. This edge structure is already visible in finite systems through the leading zero, namely the zero closest to the real axis (Peng et al., 2014, Wei et al., 2012).

For continuous phase transitions, the scaling of leading zeros yields universal critical exponents. In the Ising model, high-order cumulants of the energy or magnetization can be expressed in terms of Fisher or Lee–Yang zeros, and the imaginary part of the leading zero obeys finite-size scaling laws such as

$h$3

This provides a route to $h$4 and $h$5 without driving the system through the transition, and the method was demonstrated numerically for the two- and three-dimensional Ising model (Deger et al., 2019).

3. Mathematical characterizations and generalized Lee–Yang measures

A substantial mathematical development replaces the original Ising-specific formulation by statements about Laplace or Fourier transforms of measures. In Newman’s sense, a Lee–Yang measure is an even Borel measure whose Laplace transform has no zeros in the open right half-plane; equivalently, its Fourier transform has only real zeros. Under suitable moment conditions, this property admits necessary and sufficient characterizations in terms of Wronskians of orthogonal polynomials associated with the measure. In particular, the Fourier transform belongs to the Laguerre–Pólya class exactly when the relevant Wronskians satisfy corresponding zero-location or sign conditions (1311.0596).

The Lieb–Sokal framework shows that ferromagnetic interactions preserve Lee–Yang stability when the single-spin partition function already has the required zero geometry. Recent work sharpens that picture by showing that, in some models, the interaction itself can induce the Lee–Yang property even when the single-spin factor fails to have it. This is proved for the Blume–Capel model and for annealed site-dilute Ising models, including hierarchical interactions akin to Dyson’s hierarchical model (Kozitsky, 5 Mar 2025).

The same logic extends beyond scalar spins. For isotropic vector ferromagnets and lattice fields on $h$6, a generalized Lee–Yang property is formulated in terms of an external vector field $h$7: the partition function depends on $h$8 only through $h$9, and the zero set lies on the negative real axis in that scalar variable. For strongly isotropic single-site measures with the Lee–Yang property, this generalized statement has been established for all even spin dimensions $Z(\beta,h)=\mathrm{Tr}\,e^{-\beta H(h)}.$0, including rotor models and $Z(\beta,h)=\mathrm{Tr}\,e^{-\beta H(h)}.$1- and $Z(\beta,h)=\mathrm{Tr}\,e^{-\beta H(h)}.$2-type measures (Kozitsky, 19 Mar 2026).

4. Experimental observation through coherence and correlation functions

A long-standing obstacle was that Lee–Yang zeros occur at complex fields and so are inaccessible to direct thermodynamic tuning. That obstacle was overcome by mapping complexified parameters to real-time quantum dynamics. For a probe spin weakly coupled to a many-body system, the probe coherence can equal the bath partition function evaluated at a complex field. In one formulation,

$Z(\beta,h)=\mathrm{Tr}\,e^{-\beta H(h)}.$3

so zeros of the coherence as a function of real time are in one-to-one correspondence with Lee–Yang zeros of the partition function (Wei et al., 2012, Peng et al., 2014).

This mapping enabled the first experimental observation of Lee–Yang zeros in liquid-state NMR using trimethylphosphite. An effective ferromagnetic Ising bath of nine $Z(\beta,h)=\mathrm{Tr}\,e^{-\beta H(h)}.$4 spins was coupled to a $Z(\beta,h)=\mathrm{Tr}\,e^{-\beta H(h)}.$5 probe spin, and the zero crossings of the probe coherence identified the Lee–Yang zeros on the unit circle. The same experiment reconstructed the free energy of the bath and determined its phase transition temperature (Peng et al., 2014).

A related route uses two-time probe-spin correlation functions. For a bath Hamiltonian $Z(\beta,h)=\mathrm{Tr}\,e^{-\beta H(h)}.$6 with $Z(\beta,h)=\mathrm{Tr}\,e^{-\beta H(h)}.$7, the two-time correlator of a probe spin is proportional to the bath partition function at a complex magnetic field $Z(\beta,h)=\mathrm{Tr}\,e^{-\beta H(h)}.$8. Its zeros therefore reproduce the Lee–Yang zeros along a real-time trajectory in parameter space (Gnatenko et al., 2017).

In the thermodynamic limit, when the Lee–Yang zeros form a continuum with an edge, the probe coherence can exhibit a sudden-death phenomenon at the corresponding critical time. This gives a dynamical manifestation of Yang–Lee edge singularities and makes the time–temperature correspondence operational rather than merely formal (Wei et al., 2012).

5. Dynamical Lee–Yang property out of equilibrium

The Lee–Yang framework extends naturally to non-equilibrium trajectory ensembles. For a time-integrated observable $Z(\beta,h)=\mathrm{Tr}\,e^{-\beta H(h)}.$9, such as dynamical activity, one defines the moment generating function

$Z$0

whose zeros $Z$1 in the complex $Z$2-plane are the dynamical Lee–Yang zeros. At long times many systems satisfy $Z$3, and non-analyticities of the scaled cumulant generating function $Z$4 arise when these zeros accumulate and pinch the real axis. In this sense, the Lee–Yang property becomes a statement about phase transitions in trajectory space (Brandner et al., 2016).

For finite time, the cumulants of $Z$5 are related directly to the zeros: $Z$6 High-order cumulants are dominated by the pair of zeros closest to the origin, which makes it possible to infer leading dynamical Lee–Yang zeros from measured fluctuation data at the physical point $Z$7 (Brandner et al., 2016).

This program was realized experimentally in a stochastic process governed by Andreev tunneling between a normal-state metallic island and superconducting leads. Using cumulants of the activity up to order $Z$8, the leading dynamical Lee–Yang zeros were extracted from short-time measurements and extrapolated to convergence points

$Z$9

Those convergence points were then used to predict the long-time large-deviation statistics of the activity, with the observed crossover interpreted as a smeared dynamical phase transition rather than a sharp non-analyticity on the real axis (Brandner et al., 2016).

6. Contemporary scope, alternative geometries, and limitations

Modern numerical work has transported Lee–Yang analysis into quantum critical systems. A statistically exact method based on stochastic series expansion computes partition-function ratios at general complex fields while sampling only the zero-field ensemble. In two-dimensional quantum antiferromagnets, this was used to extract leading Lee–Yang zeros and critical exponents for the Heisenberg bilayer and the $z=e^{-\beta h}$0–$z=e^{-\beta h}$1 model; in the deconfined case it also revealed extended rings of zeros in the plane of imaginary Néel and VBS fields (D'Emidio, 2023).

Not all Lee–Yang geometries are circular. In one-dimensional classical Rydberg blockade chains with arbitrary blockade radius, the partition function in the fugacity $z=e^{-\beta h}$2 has all zeros real and negative. This excludes phase transitions in the one-dimensional classical model and provides a distinct Lee–Yang property of hard-core type rather than unit-circle type (Li et al., 2022).

Nor is the property universal. For two-component systems related to the Gaussian free field, the Villain model retains the Lee–Yang property, but complex Gaussian multiplicative chaos does not: for $z=e^{-\beta h}$3, the corresponding moment generating functions have zeros away from the imaginary axis. The obstruction is tied to tail behavior: a distribution with some stretched-exponential moment but no quadratic exponential moment cannot possess the pure-imaginary-zero property (Newman et al., 2017).

A further abstraction replaces partition functions by tensors. A tensor with binary indices defines a multivariate polynomial, and it is called a Lee–Yang tensor with radius $z=e^{-\beta h}$4 when that polynomial is nonzero on the open polydisk of radius $z=e^{-\beta h}$5. This language is stable under tensor contraction and certain quantum operations. For quantum states with Lee–Yang radius $z=e^{-\beta h}$6, quasipolynomial-size preparation circuits exist, while Hermitian operators with Lee–Yang radius $z=e^{-\beta h}$7 have a unique principal eigenvector. This suggests that $z=e^{-\beta h}$8 is a threshold separating the classical Lee–Yang regime from a stronger zero-free regime with direct complexity-theoretic consequences (Wong et al., 3 Feb 2026).

In that broad contemporary sense, the Lee–Yang property is no longer confined to the original circle theorem. It denotes a family of zero-constrained analyticity principles—equilibrium, dynamical, multivariate, and operator-valued—whose common content is that the geometry of complex zeros governs thermodynamic singularities, fluctuation structure, and, increasingly, computational complexity.