Global analytic structure of the hard-hexagon partition function per site in the complex fugacity plane: dominant eigenvalue branches beyond Baxter's solution

Baxter's exact solution of the hard hexagon model gives the partition function per site for positive real fugacity, but a 2013 finite-size study by Assis, Jacobsen, Jensen, Maillard and McCoy showed that this solution does not determine the analytic structure in the complex fugacity plane: a necklace of partition-function zeros in the left half-plane implies that at least one additional transfer-matrix eigenvalue branch becomes dominant there, and striking algebraic regularities (factorization of the characteristic polynomial and multiplicity-two resultant zeros) hold at the widths checked. The open problem is to determine the thermodynamic-limit analytic structure of the hard-hexagon partition function per site for complex fugacity: identify and prove dominant the eigenvalue branches responsible for the necklace (or prove that the region fills with zeros), prove the finite-size algebraic properties for all system widths and determine whether they follow from integrability, settle when the free energy analytically continued from the positive real axis passes through loci where the Baxter branch loses maximum modulus, and determine which of these phenomena persist in non-integrable models of the same universality class. The authors' own 2014 (hard squares) and 2017 (Ising in a magnetic field) numerical follow-ups leave these questions open, and no resolution is known in the literature.

Background

The hard hexagon model is a lattice gas on the triangular lattice in which an occupied site excludes occupation of its six nearest neighbors; its only parameter is the fugacity (activity) $z$ , each occupied site contributing a factor $z$ to the partition function. It is equivalent to the infinite-field limit of the triangular-lattice Ising antiferromagnet and is one of the few interacting two-dimensional models solved exactly. Baxter's 1980 solution gives the partition function per site in the thermodynamic limit for positive real $z$ : two branches, $\kappa_-(z)$ (low density, $0 \le z < z_c$ ) and $\kappa_+(z)$ (high density), meet at a continuous transition at $z_c = (11+5\sqrt{5})/2 \approx 11.09$ in the three-state Potts universality class (the same critical behavior as the three-state Potts ferromagnet), while $\kappa_-$ has a branch point at $z_d = -1/z_c \approx -0.0902$ , a Yang-Lee-edge-type singularity (a branch point of the kind occurring at the edge of the Lee-Yang zero distribution of a ferromagnet in an imaginary field), believed universal for purely repulsive interactions. Joyce (1988) showed that $\kappa_\pm(z)$ are algebraic functions expressible through modular functions and their Hauptmodul (a distinguished uniformizing modular function for the genus-zero modular group involved), a structure tied to the model's Yang-Baxter integrability, i.e. to the exact-solvability relations its weights satisfy.

For a finite lattice the partition function $Z_{L_v,L_h}(z)$ is a polynomial in $z$ , specified by its zeros in the complex plane (Lee-Yang-Fisher zeros). In the thermodynamic limit the free energy is analytic in regions that are limits of zero-free regions, so the accumulating locus of zeros delimits the analyticity domains of the partition function per site. In transfer-matrix formulations (the transfer matrix being the finite matrix whose successive powers build up the partition function layer by layer) $Z$ is dominated by the largest eigenvalue $\lambda_{\max}$ , and candidate accumulation loci are the equimodular curves of the transfer-matrix eigenvalues — curves on which two eigenvalues have equal modulus so that dominance can be exchanged.

Assis, Jacobsen, Jensen, Maillard and McCoy (2013) computed partition-function zeros for square lattices as large as $39 \times 39$ with cylindrical boundary conditions (up to $27 \times 27$ toroidal) and transfer-matrix eigenvalues in the translationally invariant $P = 0$ sector for widths up to $L_h = 30$ (matrix dimension 31836). They found that Baxter's branches do not describe the complex- $z$ structure: (i) a ring-shaped 'necklace' of zeros lies in the left half-plane around part of the curve where $|\kappa_-| = |\kappa_+|$ , a locus incompatible with a partition function per site built only from $\kappa_\pm$ , so at least one further eigenvalue must become dominant there; (ii) in the translationally invariant momentum sector $P = 0$ , the characteristic polynomial of the transfer matrix factors into exactly two irreducible integer-coefficient polynomials for $L_h = 12, 15, 18$ ; (iii) almost all zeros of the resultant of that characteristic polynomial (the resultant being a determinant formed from the polynomial's coefficients whose zeros locate potential singularities of the eigenvalue branches) have multiplicity two at the same widths, indicating eigenvalue crossings rather than singularities; and (iv) the earlier conjecture of Wood, Turnbull and Ball (1989) on the hard-hexagon zero locus is incorrect. The paper proposes extending Baxter's methods to the part of the complex plane not covered by $\kappa_\pm(z)$ , on the model's modular-function and Hauptmodul structure. For contrast, the non-integrable hard-square model at $z = -1$ has all transfer-matrix eigenvalues on the unit circle (roots of unity) with number-theoretic dependence on the lattice dimensions (Fendley, Schoutens and van Eerten 2005), and its resultant zeros have multiplicity one.

Problem Statement

Determine the analytic structure of the hard-hexagon partition function per site in the full complex fugacity plane, extending Baxter's positive-real-axis solution. The single scientific objective is to establish which transfer-matrix eigenvalue branch dominates each region of the complex $z$ -plane in the thermodynamic limit, and thereby the limiting locus of partition-function zeros and the analyticity domains of the free energy, together with which of the governing features are consequences of integrability. Concretely, the source work leaves open:

Necklace mechanism and fate. Prove whether the necklace of partition-function zeros in the left half-plane persists as the lattice size $L \to \infty$ ; identify the transfer-matrix eigenvalue branch(es) beyond the Baxter branches $\kappa_\pm(z)$ that become dominant (of maximum modulus) in the necklace region; determine whether the necklace region becomes partially or fully filled with zeros in the limit; and establish the algebraic or analytic origin of these extra eigenvalues.
Algebraic generality. Prove or disprove that for all system widths $L_h$ the $P=0$ characteristic polynomial of the transfer matrix factorizes over the integers (and whether always into exactly two irreducible factors), and that almost all zeros of the corresponding resultant have multiplicity two; and determine whether these properties follow from the model's integrability, as conjectured.
Analytic continuation and eigenvalue dominance. Determine whether, and under what conditions, the partition function per site analytically continued from the positive real axis can be continued through the loci where the corresponding Baxter branch ceases to be the maximum-modulus transfer-matrix eigenvalue — equivalently, when the free energy on the real axis determines analyticity throughout the complex plane. (For hard hexagons the locus on the negative real axis is the segment $z \le z_d$ .)
Universality companion. Determine whether these phenomena — the necklace, the factorizations, the multiplicity-two resultants, and the ability to continue the free energy through loci of zeros — persist in non-integrable models in the same universality class (hard squares, Ising models in a magnetic field), or are specific consequences of integrability.

These components jointly deliver the objective: (1) and (2) supply the eigenvalue branches and the finite-size algebraic structure from which the thermodynamic limit is built, (3) fixes the resulting analyticity of the free energy, and (4) identifies the scope of the mechanisms.

Each component's constituent claims are independently refutable, and the four components are jointly posed: settling a proper subset of the components, or of the claims within one component, does not settle the rest.

References

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