Hard-hexagon partition function for complex fugacity: thermodynamic-limit persistence of the zero necklace and transfer-matrix eigenvalue branches beyond Baxter's solution

The hard hexagon model, a lattice gas on the triangular lattice with nearest-neighbor exclusion, was exactly solved by Baxter for positive real fugacity $z$, where the partition function per site is given by two analytic branches $\kappa_-(z)$ (low density) and $\kappa_+(z)$ (high density). A precision finite-size study of partition-function zeros and transfer-matrix eigenvalues showed in 2013 that these two branches are not sufficient to describe the analytic structure of the partition function per site in the full complex $z$ plane: an additional 'necklace' of equimodular curves and zeros appears in the left half-plane, and the $\kappa_\pm$ equimodular curve does not bound it. The unresolved problem is to determine the analytic structure of the hard-hexagon partition function per site for complex fugacity in the thermodynamic limit: to prove or disprove that the necklace survives $L\to\infty$ (and whether it becomes partially or fully filled with zeros), to identify the transfer-matrix eigenvalue branches beyond $\kappa_\pm$ that dominate in the necklace region, to prove or refute for all system widths the observed algebraic regularities (factorization of the $P=0$ characteristic polynomial into two irreducible factors, and multiplicity-two zeros of the resultant), and to establish whether the free energy analytically continued from the positive real axis through loci where the Baxter branch loses maximum modulus yields the physical free energy. Follow-up studies by the same authors (2014, 2017) treat these questions only numerically or in adjacent settings; the analytic core remains open.

Background

The hard hexagon model is a lattice gas of particles on the triangular lattice that exclude each other from nearest-neighbor sites; its grand partition function $Z(z)$ is a polynomial in the fugacity $z$ (the Boltzmann weight per particle). Baxter (1980) solved it exactly for positive real $z$ by exploiting the Yang-Baxter equation (the consistency condition that makes the model integrable): on a cylinder of circumference $L_h$, the partition function is generated by a transfer matrix $T(z;L_h)$, and the partition function per site is $\kappa(z)=\lim_{L_h\to\infty}\lambda_{\max}(z;L_h)^{1/L_h}$, where $\lambda_{\max}$ is the largest eigenvalue. Baxter found two branches: a low-density branch $\kappa_-(z)$, analytic in the plane cut along the real axis from $z_d=(11-5\sqrt5)/2=-0.09017\ldots$ to $-\infty$ and from $z_c=(11+5\sqrt5)/2=11.09017\ldots$ to $+\infty$, and a high-density branch $\kappa_+(z)$, cut from $z_c$ to $-\infty$; the two meet in a first-order transition at $z_c$. Joyce (1988) and Tracy, Grove and Newman (1987) showed that this solution has a remarkable modular structure: the branches are algebraic functions expressible through a Hauptmodul (the generator of the function field of a genus-zero modular curve), and $\kappa_+$ satisfies an explicit algebraic equation.

For a finite $L_v\times L_h$ lattice $Z(z)$ is a polynomial, so it is specified by its zeros; following Lee and Yang, one studies where these zeros condense as $L\to\infty$, since the free energy is analytic in the regions that are the limits of the zero-free regions. On a cylinder the relevant objects are equimodular curves: loci in the complex $z$ plane where two transfer-matrix eigenvalues have equal modulus; when the thermodynamic limit commutes with the eigenvalue limit, the limiting zeros lie on the limiting equimodular curves, and the phase of the dominant eigenvalue determines the density of zeros along the curve. Wood (1985) and Wood, Turnbull and Ball (1989) computed hard-hexagon zeros on small lattices and conjectured a simple high-density equimodular characterization of the zero locus.

Assis, Jacobsen, Jensen, Maillard and McCoy (J. Phys. A 46 (2013) 445202) computed partition-function zeros on lattices up to $39\times39$ and transfer-matrix eigenvalues up to circumference $L_h=30$ (dimension 31836 in the translationally invariant sector of momentum $P=0$). They found: (i) the Wood-Turnbull-Ball conjecture is incorrect and must be replaced by equimodular conditions involving both $\kappa_+$ and $\kappa_-$; (ii) in the left half of the complex $z$ plane an additional structure appears that they call the necklace (reference [1]): the system of regions of the left half-plane carved out by the equimodular curves of the largest transfer-matrix eigenvalues in the translationally invariant $P=0$ sector, which is incompatible with a partition function built only from $\kappa_\pm$ — the $\kappa_\pm$ equimodular curve, which crosses the negative real axis at $z=-5.9425104\ldots$, splits the necklace into parts instead of bounding it; (iii) partition-function zeros enter the interior of the necklace region only from lattice size $30\times30$ upward (there are none there for $24\times24$ and $27\times27$); (iv) in the $P=0$ sector the characteristic polynomial of the transfer matrix factors into two irreducible polynomials with integer coefficients for $L_h=12,15,18$, and almost all zeros of the resultant of that polynomial have multiplicity two, indicating level crossings rather than singularities — both properties presumed to follow from integrability, but with no such theorem in the literature; (v) they also obtained a new algebraic equation for the low-density branch $\kappa_-(z)$. They note that there is no general theorem stating when the free energy of a system can be continued through the locus of its partition-function zeros, so the relation between the exactly known real-axis solution and the analytic structure elsewhere in the complex plane is open.

Problem Statement

Determine the analytic structure of the hard-hexagon partition function per site $\kappa(z)$ in the full complex fugacity plane in the thermodynamic limit, completing beyond the positive real axis what Baxter's solution and its modular (Hauptmodul) refinements give on it. Concretely, the source asks for the following jointly supporting results.

1. Necklace in the thermodynamic limit: a proof or disproof that the necklace of equimodular curves and partition-function zeros in the left half of the complex $z$ plane persists as $L\to\infty$, together with a determination of whether the necklace region becomes partially or fully filled with zeros in the limit.

2. Dominant branches and their mechanism: identification of which transfer-matrix eigenvalue branch(es) beyond the Baxter branches $\kappa_\pm(z)$ become dominant in the necklace region as $L_h\to\infty$ — the mechanism proposed in section 6.2 of the source as the simplest account of the necklace behavior, whose verification, as the source notes, requires further analytic computation.

3. Algebraic regularities for all widths: a proof, or a refutation, of the finite-size observations for all circumferences $L_h$ — the factorization of the $P=0$ characteristic polynomial of the transfer matrix into two irreducible polynomials with integer coefficients (and whether a factorization into more than two factors can occur for larger widths), and the property that almost all zeros of the resultant of that polynomial have multiplicity two — together with a determination of whether these properties follow from the integrability of the model.

4. Analytic continuation through loss of dominance: a criterion, established at least for hard hexagons, for whether the partition function per site analytically continued from the positive real axis can be continued through 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 the analyticity of the free energy in the rest of the complex plane, given that no general theorem on continuation through a locus of zeros is known.

These four deliverables are jointly required: a resolution of the task must address each of them, and a proof or disproof of any single item is an independently meaningful partial result that does not by itself complete the objective. The common objective of these deliverables is a complete analytic description of $\kappa(z)$ over the complex plane: which eigenvalue branches control each region, which loci carry the limiting zeros, and how far the exactly known real-axis solution extends. The companion questions the same source raises for non-integrable relatives (hard squares, Ising models in finite magnetic field) were deliberately deferred by its authors to separate studies and are recorded as adjacent open problems, not as part of this problem's completion boundary.

References

1. M. Assis, J. L. Jacobsen, I. Jensen, J-M. Maillard, B. M. McCoy, The hard hexagon partition function for complex fugacity, J. Phys. A: Math. Theor. 46 (2013) 445202. DOI: 10.1088/1751-8113/46/44/445202; arXiv:1306.6389; https://arxiv.org/abs/1306.6389
2. M. Assis, J. L. Jacobsen, I. Jensen, J-M. Maillard, B. M. McCoy, Integrability versus non-integrability: hard hexagons and hard squares compared, J. Phys. A: Math. Theor. 47 (2014) 445001. DOI: 10.1088/1751-8113/47/44/445001; arXiv:1406.5566; https://arxiv.org/abs/1406.5566
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