Analytic structure of the hard-hexagon partition function per site in the complex fugacity plane: dominant transfer-matrix eigenvalue branches, the necklace region and its zeros, and the algebraic origin of the extra eigenvalues

Baxter's exact solution of the hard hexagon model gives the partition function per site $\kappa_{\pm}(z)$ on the positive real fugacity axis, where the model has a disorder-order transition at $z_c=(11+5\sqrt5)/2$. A precision finite-size study by Assis, Jacobsen, Jensen, Maillard and McCoy (J. Phys. A 46, 445202 (2013)) showed that these two branches do not determine the analyticity of the partition function per site in the full complex $z$ plane: zeros and transfer-matrix equimodular curves reveal a 'necklace' of regions in the left half-plane, extending on the negative real axis to a point that extrapolates to $z\approx-9$ to $-10$, well beyond the crossing $z=-5.9425\ldots$ of the $|\kappa_+|=|\kappa_-|$ curve, so at least one additional transfer-matrix eigenvalue must become dominant there. The open problem is to determine this structure analytically: identify and prove dominance of the eigenvalue branch(es) that govern the necklace region in the thermodynamic limit and settle whether the necklace is partially or fully filled with partition-function zeros; characterize the algebraic/modular (Hauptmodul) origin of these extra eigenvalues, including proving or disproving that the observed finite-size regularities (factorization of the $P=0$ characteristic polynomial into two irreducible integer-coefficient factors; multiplicity two of all but a fixed number of resultant zeros) hold for all lattice widths as consequences of integrability; and determine on this basis the piecewise-analytic form of the partition function per site over the complex plane and the maximal domain to which it continues the real-axis free energy. No resolution of this problem is known in the literature to date.

Background

The hard hexagon model is a lattice gas of hard particles on the triangular lattice with nearest-neighbor exclusion: each occupied site forbids occupation of its six neighbors. Its grand partition function $\Xi(z)=\sum_N Z_N z^N$, where $Z_N$ counts the allowed configurations with $N$ particles, is a polynomial in the fugacity $z$ (the thermodynamic activity, the weight carried by each particle), and for a finite $L_h\times L_v$ lattice it can be written as a trace of a power of a transfer matrix acting on row states. Baxter solved the model exactly in 1980 (J. Phys. A 13, L61) using corner transfer matrices (transfer-matrix factors built on quarter-lattices rather than horizontal strips), exploiting the fact that the model satisfies the Yang-Baxter equation (the solvability condition for two-dimensional exactly solved models). In the thermodynamic limit on the positive real axis the partition function per site $\kappa(z)$ is $\kappa_-(z)$ for $z\le z_c$ and $\kappa_+(z)$ for $z\ge z_c$, with the disorder-order transition at $z_c=(11+5\sqrt5)/2\approx11.09$ in the three-state Potts universality class (the critical behavior of the $q=3$-state Potts ferromagnet). Joyce, and Tracy, Grove and Newman, showed that $\kappa_{\pm}(z)$ are algebraic functions tied to a remarkable modular structure: they are expressible through modular functions (special functions of a complex parameter transforming covariantly under the modular group of $2\times2$ integer matrices) whose generating parameter is related to the icosahedral equation (the algebraic relation invariant under the rotational symmetry group of the icosahedron), with a Hauptmodul (the generator of the function field of the underlying modular curve) playing the central role.

For complex fugacity, Yang-Lee theory applies: phase boundaries of the thermodynamic limit arise as loci where the zeros of the finite-system partition functions accumulate. For real positive $z$ the dominant (largest-modulus) transfer-matrix eigenvalue is unique by Perron-Frobenius theory (the spectral theory of matrices with non-negative entries), but for complex $z$ several eigenvalues can have equal modulus, and the loci where this happens (equimodular curves) organize the finite-size spectra. Zeros of the hard hexagon partition function lie on the negative real axis for $z\le z_d$, where $z_d=(11-5\sqrt5)/2=-1/z_c\approx-0.0902$; the endpoint $z_d$ carries the Yang-Lee edge singularity, an edge exponent $-1/6$ consistent with Cardy's conformal field theory of the edge, and the density of zeros diverges there as $(1-z/z_d)^{-1/6}$. This singular behavior is believed to be universal for systems with purely repulsive interactions (the repulsive-core singularity, since identified with Yang-Lee edge criticality).

In 2013 Assis, Jacobsen, Jensen, Maillard and McCoy made a precision finite-size study (partition-function zeros on lattices up to $39\times39$; transfer-matrix eigenvalues in the zero-momentum ($P=0$) sector up to width $L_h=30$, matrix dimension 31836). Away from the left half-plane the finite-size equimodular curves converge on the curve $|\kappa_+(z)|=|\kappa_-(z)|$, confirming Baxter's branches; but in the left half-plane the eigenvalue crossings form a chain of regions they called a 'necklace', whose leftmost crossing on the negative real axis extrapolates (in $1/L_h$) to a value between 9 and 10 in magnitude, far to the left of the $\kappa_{\pm}$ equimodular crossing at $z=-5.9425\ldots$. Hence at least one additional eigenvalue must become dominant in the necklace region as $L_h\to\infty$, and the two functions $\kappa_{\pm}(z)$, which are singular only at $z_d$, $z_c$ and infinity, cannot describe the analyticity of the partition function per site in the complex plane. The same study refuted the 1989 conjecture of Wood, Turnbull and Ball on the hard hexagon zero locus, derived a new algebraic equation for $\kappa_-(z)$, and found two non-generic algebraic regularities at accessible widths: for $L_h=12,15,18$ the characteristic polynomial of the transfer matrix in the translationally invariant $P=0$ sector factors into a product of exactly two irreducible polynomials with integer coefficients (hard squares show no such factorization), and all but a fixed number of zeros of the resultant of that characteristic polynomial (the algebraic eliminant whose roots locate potential branch points of the eigenvalue branches) have multiplicity two, indicating level crossings rather than singularities (the hard square resultant roots have multiplicity one). For toroidal boundary conditions the authors also conjectured, from the observed squeezing of the $P=\pm2\pi/3$ sectors, that only the $P=0$ sector survives as $L_h\to\infty$ except on the equimodular curves themselves.

What is missing is analytic understanding of all of these observations. The finite-size data strongly suggest the necklace survives $L_h\to\infty$ and that one or more specific additional eigenvalue branches become dominant there, but no analytic demonstration exists; no extension of Baxter's method accounts for the extra eigenvalues or ties them to the modular/Hauptmodul structure; the factorization and multiplicity-two properties have been checked only at $L_h=12,15,18$ and no theorem deriving them from integrability is known ('no such theorem is in the literature', in the authors' words); and it is unknown in general when the free energy obtained on the real axis determines the analyticity of the partition function per site throughout the complex plane, beyond the region where it corresponds to the maximum-modulus eigenvalue.

Problem Statement

Determine, analytically and in the thermodynamic limit, the structure of the hard hexagon partition function per site throughout the complex fugacity plane, completing Baxter's exact solution beyond the positive real axis. Concretely, establish the following jointly linked results for the hard hexagon model (triangular lattice, nearest-neighbor exclusion, transfer matrix of width $L_h$ with periodicity compatible with the three-sublattice structure: the ordered phase occupies one of three interpenetrating sublattices, and the momentum sectors of the transfer matrix relevant to the dominant eigenvalues are $P=0$ and $P=\pm2\pi/3$):

(i) Identify the transfer-matrix eigenvalue branch(es), beyond the two Baxter branches whose per-site limits are $\kappa_\pm(z)$, that become dominant in the necklace region of the left half $z$-plane as $L_h\to\infty$, and prove their dominance. (The source conjectures, without establishing it, that only the $P=0$ momentum sector survives in the thermodynamic limit, which would make the necklace common to cylindrical and toroidal boundary conditions.) On this basis settle whether the thermodynamic-limit necklace locus is partially or fully filled with zeros of the partition function.

(ii) Characterize the algebraic/analytic origin of these additional eigenvalues. The source proposes, as one route toward this, an extension of Baxter's method tying the extra eigenvalues to the modular/Hauptmodul structure of the model (the icosahedrally related modular functions through which $\kappa_\pm$ are expressed); its explicit open item is the analytic verification of the extra-eigenvalue mechanism itself, so any analytic construction that derives the extra eigenvalues from the integrability of the model addresses this item. As algebraic fingerprints of the same structure, prove or disprove that for every width $L_h$ compatible with the three-sublattice periodicity (that is, $L_h$ a multiple of three, the widths considered in the source) the characteristic polynomial of the transfer matrix in the translationally invariant $P=0$ sector factors into a product of exactly two irreducible polynomials with integer coefficients, and that all zeros of its resultant have multiplicity two except for at most a fixed number of them, independent of $L_h$, and derive these properties (or their failure) from the integrability of the model.

(iii) Determine the piecewise-analytic form of the partition function per site over the whole complex $z$ plane - which algebraic branch governs each region bounded by the thermodynamic limits of the equimodular curves - and the maximal domain to which this function is the analytic continuation of the free energy on the positive real axis; in particular, whether and how it can be continued through the loci where the Baxter branch $\kappa_\pm$ ceases to be the maximum-modulus eigenvalue.

Deliverable (i) without (ii) and (iii) would identify the dominant branches without explaining them; (ii) supplies the algebraic framework that makes (i) and (iii) derivable rather than conjectural; (iii) is the completion of the objective, of which (i) and (ii) are the necessary mechanisms. A solution that rigorously refutes a load-bearing part also counts as an answer. The independently refutable parts are the two-factor factorization of the $P=0$ characteristic polynomial for all widths $L_h$ (refuted by a counterexample at a single admissible width) and the multiplicity-two property of all but a fixed number of resultant zeros for all widths (refuted by showing that the number of resultant zeros of multiplicity other than two grows without bound with $L_h$), and the survival of the necklace in the thermodynamic limit and the conjectured survival of only the $P=0$ momentum sector (thermodynamic-limit claims, each to be established for the limit rather than merely at finite size). A refutation of any one of these parts, established for the model itself, resolves that part and counts as an answer, leaving the remaining parts open.

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://doi.org/10.1088/1751-8113/46/44/445202
2. R. J. Baxter, "Hard hexagons: exact solution", J. Phys. A: Math. Gen. 13 (1980) L61-L70. DOI: 10.1088/0305-4470/13/3/007. https://doi.org/10.1088/0305-4470/13/3/007
3. R. J. Baxter, P. A. Pearce, "Hard hexagons: interfacial tension and correlation length", J. Phys. A: Math. Gen. 15 (1982) 897-910. DOI: 10.1088/0305-4470/15/3/027. https://doi.org/10.1088/0305-4470/15/3/027
4. G. S. Joyce, "On the hard-hexagon model and the theory of modular functions", Phil. Trans. R. Soc. Lond. A 325 (1988) 643-702. DOI: 10.1098/rsta.1988.0077. https://doi.org/10.1098/rsta.1988.0077
5. C. A. Tracy, L. Grove, M. F. Newman, "Modular properties of the hard hexagon model", J. Stat. Phys. 48 (1987) 477-502. DOI: 10.1007/BF01019683. https://doi.org/10.1007/BF01019683
6. D. W. Wood, R. W. Turnbull, J. K. Ball, "An observation on the partition function zeros of the hard hexagon model", J. Phys. A: Math. Gen. 22 (1989) L105-L109. DOI: 10.1088/0305-4470/22/3/008. https://doi.org/10.1088/0305-4470/22/3/008
7. J. L. Cardy, "Conformal invariance and the Yang-Lee edge singularity in two dimensions", Phys. Rev. Lett. 54 (1985) 1354-1356. DOI: 10.1103/PhysRevLett.54.1354. https://doi.org/10.1103/PhysRevLett.54.1354
8. S.-N. Lai, M. E. Fisher, "The universal repulsive-core singularity and Yang-Lee edge criticality", J. Chem. Phys. 103 (1995) 8144-8155. DOI: 10.1063/1.470178. https://doi.org/10.1063/1.470178
9. Y. Park, M. E. Fisher, "Identity of the universal repulsive-core singularity with Yang-Lee edge criticality", Phys. Rev. E 60 (1999) 6323-6328. DOI: 10.1103/PhysRevE.60.6323. https://doi.org/10.1103/PhysRevE.60.6323
10. 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://doi.org/10.1088/1751-8113/47/44/445001
11. M. Assis, J. L. Jacobsen, I. Jensen, J.-M. Maillard, B. M. McCoy, "Analyticity of the Ising susceptibility: an interpretation", J. Phys. A: Math. Theor. 50 (2017) 365203. DOI: 10.1088/1751-8121/aa8128; arXiv:1705.02541. https://doi.org/10.1088/1751-8121/aa8128
12. A. García-Saez, T.-C. Wei, "Density of Yang-Lee zeros in the thermodynamic limit from tensor network methods", Phys. Rev. B 92 (2015) 125132. DOI: 10.1103/PhysRevB.92.125132; arXiv:1508.07751. https://doi.org/10.1103/PhysRevB.92.125132
13. A. Haldar, O. Tavakol, H. Ma, T. Scaffidi, "Hidden critical points in the two-dimensional O(n>2) model: exact numerical study of a complex conformal field theory", Phys. Rev. Lett. 131 (2023) 131601. DOI: 10.1103/PhysRevLett.131.131601; arXiv:2303.02171. https://doi.org/10.1103/PhysRevLett.131.131601
14. M. V. Tamm, M. Dudka, N. Pospelov, G. Oshanin, S. Nechaev, "From steady-state TASEP model with open boundaries to 1D Ising model at negative fugacity", J. Stat. Mech. (2022) 033201. DOI: 10.1088/1742-5468/ac52a5; arXiv:2112.04377. https://doi.org/10.1088/1742-5468/ac52a5
15. S. Sarma, "A numerical study of the zeros of the grand partition function of k-mers on strips of width k", J. Phys. A: Math. Theor. 58 (2025) 165001. DOI: 10.1088/1751-8121/adc9e8; arXiv:2409.07744. https://doi.org/10.1088/1751-8121/adc9e8
16. H. H. Otto, "Phase transitions governed by the fifth power of the golden mean and beyond", World J. Condens. Matter Phys. 10 (2020) 135-158. DOI: 10.4236/wjcmp.2020.103009. https://doi.org/10.4236/wjcmp.2020.103009