Partition Function Zeros in Statistical Physics

• Partition function zeros are the complex roots of a finite-volume partition function that serve as precise indicators of phase transitions.
• Their global distribution, scaling behavior, and geometric loci elucidate the order and universality of both classical and quantum transitions.
• Computational methods such as combinatorial enumeration, Monte Carlo simulations, and integral representations enable practical determination of these zeros.

Partition function zeros are complex solutions to the condition $Z(\boldsymbol{\lambda})=0$, where $Z$ is the finite-volume partition function of a statistical mechanical model analytically continued in one or more thermodynamic or coupling variables $\boldsymbol{\lambda}$ (e.g., temperature, fugacity, magnetic field, crystal field). First conceived by Yang and Lee and then Fisher for the purposes of rigorously characterizing phase transitions, the study of partition function zeros underlies a universal paradigm for detecting and classifying singularities, criticalities, and non-analytic features of the free energy in both classical and quantum many-body systems. The global distribution, scaling behavior, and geometric loci of these zeros collectively encode not only the existence but the precise order and universality class of transitions, and, in disordered or frustrated systems, also diagnose the onset of phenomena such as spin-glass chaos or condensation transitions of arbitrary order.

1. Foundational Definitions and General Principles

For a finite system, the partition function $Z(\lambda)$—whether canonical, grand-canonical, or extended—is a polynomial (or entire function) in the relevant parameter $\lambda$ (such as $z=e^{\beta\mu}$, $y=e^{\beta\epsilon}$, $t=e^{-2\beta J}$, or $e^{\beta\Delta}$). Its zeros in the complex plane, called partition function zeros, come in complex-conjugate pairs and are physically inaccessible for any real parameter value, as $Z$ is strictly positive for real $Z$0. However, in the thermodynamic limit, curves or regions of zeros can approach ("pinch") the real axis, thereby causing non-analyticities in the free energy,

$Z$1

which signal phase transitions (Dijk et al., 2013).

Two principal classes are:

• Lee–Yang zeros: Zeros in the complex fugacity $Z$2 or complex field $Z$3 plane at fixed real temperature (originally for the grand-canonical ensemble).
• Fisher zeros: Zeros in the complex inverse-temperature $Z$4 (or $Z$5) plane at fixed particle number or other parameters (canonical or semi-canonical ensemble).

The accumulation locus, pinch geometry, and local density of zeros near the real axis collectively determine:

• Existence and location of phase transitions,
• Order of the transition (discontinuous/first-order vs. continuous/second-order/tricritical),
• Universality class and critical exponents,
• System-specific features such as spin-glass order, frustration, or condensation (Takahashi, 2011, Krasnytska et al., 2015, Bialas et al., 2023, Honchar et al., 8 Jan 2025).

2. Mathematical Structure and Distribution of Zeros

Let $Z$6 be a partition function of degree $Z$7. It can be factorized as

$Z$8

where the set $Z$9 defines the partition function zeros in the complex plane (Dijk et al., 2013). For canonical partition functions as a function of $\boldsymbol{\lambda}$0, these are called Fisher zeros; as a function of fugacity, Lee–Yang zeros. Classes of models, including Ising variants, Potts models, polymers, quantum systems, or urn models, lead to partition functions which, through combinatorial or algebraic enumeration, yield explicit polynomials whose zeros are amenable to direct calculation up to substantial sizes (Lee et al., 2011, Bialas et al., 2023, Beaton et al., 2017).

As $\boldsymbol{\lambda}$1, the zeros form smooth curves or regions in the complex plane. Critical singularities on the real axis manifest when this locus pinches the real line. The local density $\boldsymbol{\lambda}$2 of zeros near a critical point determines the order: finite $\boldsymbol{\lambda}$3 (first order), $\boldsymbol{\lambda}$4 for $\boldsymbol{\lambda}$5 (continuous of order $\boldsymbol{\lambda}$6) (Majumdar et al., 2019, Bialas et al., 2023).

Table: Key Zero Types and Their Roles

Type of Zero	Complex Plane	Physically Signals
Lee–Yang zero	$\boldsymbol{\lambda}$7 or $\boldsymbol{\lambda}$8	Field-driven or condensation
Fisher zero	$\boldsymbol{\lambda}$9 or $Z(\lambda)$0	Thermal (T-driven) transitions
Crystal-field zero	$Z(\lambda)$1	Anisotropy/tricriticality

3. Physical Interpretation: Phase Transitions and Criticality

The analytic structure of zeros encodes phase behavior. For first-order transitions, the density of zeros is non-vanishing at the critical point; zeros approach the real axis at a nonzero angle (often vertically), producing a discontinuity in the first derivative of $Z(\lambda)$2. For continuous transitions, the density vanishes as a power law, $Z(\lambda)$3; the pinch occurs at a finite angle, and the free energy develops a branch-point singularity (Majumdar et al., 2019, Dijk et al., 2013, Honchar et al., 8 Jan 2025, Bialas et al., 2023).

Key correspondences:

• First-order: zeros approach the real axis vertically, with nonzero density; coexistence region or latent heat (Aslyamov et al., 2018, Bialas et al., 2023).
• Continuous (second or higher order): zeros approach at nonzero angle, density vanishes as $Z(\lambda)$4, exponent $Z(\lambda)$5 related to critical exponents such as $Z(\lambda)$6 (Majumdar et al., 2019, Krasnytska et al., 2015).
• Multicritical/tricritical: scaling of zeros and impact angle change to reflect crossover exponents and universality class (Honchar et al., 8 Jan 2025).

This framework is universal, applicable to both classical and quantum models, disordered or frustrated systems, models with continuous symmetry breaking, and various ensemble choices (Matsuda et al., 2010, Aslyamov et al., 2018, Shastry, 2 Apr 2025). In spin glasses, for example, the two-dimensional support of zeros in the complex parameter plane is characteristic of chaotic and nonergodic thermodynamics (Matsuda et al., 2010, Takahashi, 2011).

4. Computation and Methodologies

Several approaches enable calculation or estimation of partition function zeros:

• Combinatorial enumeration: For finite $Z(\lambda)$7, combinatorial counts of configurations as a function of energy, magnetization, or other quantities result in polynomials (e.g., $Z(\lambda)$8 for polymers), whose complex roots are found via standard algebraic methods (Lee et al., 2011, Lee et al., 2012, Beaton et al., 2017).
• Population dynamics and cavity method: In disordered or Bethe-lattice models, recursive distributional equations for cavity fields provide the distribution of local fields, whose singularities correspond to zeros (Matsuda et al., 2010).
• Integral (Hubbard-Stratonovich) representations: Mean-field and network models reduce to one-dimensional integrals, enabling an analytic or numerical determination of scaling and locus of zeros (Krasnytska et al., 2015).
• Monte Carlo/cumulant methods: The leading zeros can be efficiently extracted from high-order cumulants of energy or magnetization, avoiding explicit root finding (Majumdar et al., 2019, Gessert et al., 2024, Sarkanych et al., 2021).
• Graph-theoretic contraction and zero-free regions: Techniques using Grace–Szegő–Walsh polarization and Asano–Ruelle contraction yield explicit zero-free domains and deterministic approximation algorithms (Guo et al., 2019).
• Boundary condition engineering: Special boundary choices (e.g., Brascamp-Kunz, B-K type in free-fermion models) permit partition function expressions whose zeros can be exactly enumerated in product form, establishing their locus even at finite size (Li et al., 29 Jul 2025).

5. Scaling Laws, Critical Exponents, and Universality

The asymptotic scaling of the imaginary part of the first zero close to the real axis provides direct access to critical exponents: $Z(\lambda)$9 where $\lambda$0 is typically determined by the universality class and critical exponents ($\lambda$1, $\lambda$2, $\lambda$3, $\lambda$4) (Lee et al., 2011, Honchar et al., 8 Jan 2025).

For example, the scaling of Fisher and Lee–Yang zeros near critical points—such as in the Ising and Blume–Capel models—obeys

$\lambda$5

and the complex-plane impact angle is related to exponents and amplitude ratios via

$\lambda$6

(Krasnytska et al., 2015, Honchar et al., 8 Jan 2025).

Table: Partition Function Zeros and Critical Exponents

Transition Type	Zero Density at Pinch	Zero Scaling	Exponent Relation
First-order	$\lambda$7	$\lambda$8	$\lambda$9
Continuous	$z=e^{\beta\mu}$0	$z=e^{\beta\mu}$1	$z=e^{\beta\mu}$2
Higher order	$z=e^{\beta\mu}$3	$z=e^{\beta\mu}$4	$z=e^{\beta\mu}$5 related to divergence of $z=e^{\beta\mu}$6

6. Models Illustrating Zeros Structure

Numerous specific models have been analyzed:

• Polymers (collapse, adsorption): Exact enumeration up to moderate $z=e^{\beta\mu}$7 confirms scaling of first zero and yields precise crossover exponents, with universality checks for $z=e^{\beta\mu}$8-transition in 2D and 3D (Lee et al., 2011, Lee et al., 2012, Beaton et al., 2017, Rensburg, 2016).
• Spin glasses (Bethe lattice, mean-field): In the $z=e^{\beta\mu}$9 model and Random Energy Model (REM), zeros map out full paramagnetic, ferromagnetic, and spin-glass phases. Two-dimensional support of zeros is linked to chaotic response and replica symmetry breaking, with nonanalyticities not restricted to isolated points (Matsuda et al., 2010, Takahashi, 2011).
• Network models: In Ising models on annealed scale-free networks, the critical exponents and impact angles become $y=e^{\beta\epsilon}$0-dependent; the Lee–Yang circle theorem fails for $y=e^{\beta\epsilon}$1 (Krasnytska et al., 2015).
• Quantum systems: Zeros can be characterized by continuous self-energy equations (e.g., Hubbard model, lattice fermions), with the zeros of the grand partition function directly mapped to solutions involving the self-energy at specific virtual energies (Shastry, 2 Apr 2025).
• NPT ensemble and fluids: Analogues of Lee–Yang and Fisher zeros rigorously apply even in the isothermal-isobaric (NPT) ensemble, with exact or approximate Szegő curves signaling condensation or criticality in fluids such as the Tonks gas or van der Waals fluid (Aslyamov et al., 2018).
• Multiple fields: Models such as the Blume-Capel include additional parameters (e.g., crystal field), and zeros in the complex-$y=e^{\beta\epsilon}$2 plane distinguish tricritical crossovers and anisotropy-driven transitions (Honchar et al., 8 Jan 2025).
• Antiferromagnets and trees: For the antiferromagnetic Ising model on Cayley trees and bounded-degree graphs, dynamical systems theory precisely classifies the density and locus of Lee–Yang zeros on the unit circle (Bencs et al., 2019).

7. Applications, Extensions, and Broader Impact

Partition function zeros have been systematically utilized to:

• Precisely determine phase transition locations, including systems with strong corrections to scaling or non-standard universality values (Gessert et al., 2024),
• Distinguish transition order and crossover regimes (critical vs. tricritical/scaling crossover) in models exhibiting complex phase diagrams (Honchar et al., 8 Jan 2025),
• Inform the design and guarantees of approximation algorithms based on zero-free regions of the partition function (e.g., Barvinok's polynomial interpolation and FPTAS for counting problems) (Guo et al., 2019),
• Diagnose breakdown or generalizations of standard theorems (e.g., Lee–Yang circle theorem on nontrivial graphs or with disorder) (Krasnytska et al., 2015, Bencs et al., 2019).

Moreover, the zeros framework is used in quantum many-body theory—connecting the distribution of complex zeros to self-energy properties and the onset of quantum criticality or pairing (Shastry, 2 Apr 2025).

Partition function zeros thus provide a rigorous, unifying, and highly discriminating set of tools for both theoretical characterization and practical computation of critical phenomena across classical, quantum, ordered, disordered, and geometrically complex statistical models.