Thermodynamic-limit analytic structure of the hard-hexagon partition function at complex fugacity: the necklace of zeros and the transfer-matrix eigenvalue branches beyond Baxter's solution

Baxter's exact solution of the hard hexagon model — the nearest-neighbour-exclusion lattice gas on the triangular lattice — gives the grand partition function per site for real positive fugacity $z$ as two analytic branches $\kappa_{\pm}(z)$ meeting at $z_c=(11+5\sqrt{5})/2$, with the underlying modular-function (Hauptmodul) structure uncovered by Joyce and by Tracy, Grove and Newman. Assis, Jacobsen, Jensen, Maillard and McCoy (J. Phys. A 46, 445202 (2013)) showed by precision finite-size computations that these two branches do not describe the analyticity at complex fugacity: in the left half of the $z$-plane the finite-size equimodular curves of the transfer matrix form a 'necklace' of level-crossing regions whose existence requires at least one further dominant eigenvalue branch, partition-function zeros begin to enter it, the characteristic polynomial in the translationally invariant sector factorizes into two irreducible factors, and the negative-axis resultant zeros have multiplicity two. The open problem is to determine this structure analytically in the thermodynamic limit: whether the necklace persists and how it fills with zeros, which extra eigenvalue branches dominate it and their modular origin, whether the observed algebraic regularities hold for all lattice widths and follow from integrability, and when the free energy analytically continued from the positive real axis can be continued through the loci where the Baxter branch loses maximum modulus — including whether this continuation property survives in non-integrable relatives such as hard squares and the Ising model in a field.

Background

The hard hexagon model is a lattice gas of particles on the triangular lattice with hard-core exclusion between nearest-neighbour sites, each particle weighted by a fugacity $z$; its grand partition function $Z(z)$ is the independence polynomial of the triangular lattice. It is the archetype of an exactly solved interacting lattice gas: Baxter (1980) computed the thermodynamic limit of the partition function per site for real $z>0$, finding two analytic branches — the low-density branch $\kappa_{-}(z)$ for $0\le z<z_c$ and the high-density branch $\kappa_{+}(z)$ for $z_c<z<\infty$ — meeting at a second-order critical point $z_c=(11+5\sqrt{5})/2\approx 11.0902$, with a companion singular point at $z_d=-1/z_c\approx -0.0902$ on the negative real axis. The model is integrable: it satisfies the Yang-Baxter equation, a functional identity that yields a one-parameter family of mutually commuting transfer matrices and underlies the exact solvability used throughout what follows. Joyce (1988) and Tracy, Grove and Newman (1987) showed that both branches are expressible through modular functions on a genus-zero curve with icosahedral symmetry, i.e. through a Hauptmodul (a single generator of the function field of the modular curve), which makes hard hexagons a rare model where global analytic information beyond series expansions is available.

For complex fugacity, information comes from the zeros of $Z$ (the Yang–Lee zeros and their analogues): in finite volume $Z$ is a polynomial whose zeros migrate with system size, and in the thermodynamic limit the free energy is non-analytic precisely where the zeros accumulate. Equivalently, for the transfer matrix $T(z)$ connecting two lattice rows, $\lim Z^{1/N}(z)$ is governed by the eigenvalue of largest modulus, so the non-analytic loci are the 'equimodular' (level-crossing) curves on which two or more eigenvalues share the largest modulus. Baxter's solution supplies two such branches $\kappa_{\pm}(z)$; the curve $|\kappa_{+}(z)|=|\kappa_{-}(z)|$ was compared with early zero computations, and Wood, Turnbull and Ball (1989) conjectured a simple zero locus which Joyce (1989) related to the icosahedral equation.

Assis, Jacobsen, Jensen, Maillard and McCoy (2013) performed a precision finite-size study (transfer matrices for infinite strips of width $L_h$ up to $L_h=30$, with transfer-matrix dimension 31836 in the translationally invariant sector, partition-function zeros to $39\times 39$, cylindrical and toroidal boundary conditions) and found: (i) outside the left half-plane the finite-size equimodular curves converge to the $|\kappa_{+}|=|\kappa_{-}|$ curve, but beyond a 'Y-branching' point they form a chain of level-crossing regions — the 'necklace' — whose leftmost negative-axis crossing extrapolates to between $-9$ and $-10$, to the left of $z=-5.9425\ldots$ where the $\kappa_{\pm}$ curve meets the axis, implying that the necklace persists as $L_h\to\infty$ and that at least one further transfer-matrix eigenvalue branch becomes dominant there; (ii) partition-function zeros enter the necklace region for cylindrical lattices $30\times 30$ and larger, while toroidal $24\times 24$ and $27\times 27$ lattices show none (attributed to delayed finite-size effects); (iii) in the translationally invariant ($P=0$) sector — the momentum-zero subspace selected by periodic boundary conditions in the transfer direction — the characteristic polynomial of $T(z)$ factorizes into two irreducible polynomials with integer coefficients for $L_h=12,15,18$, whereas for the non-integrable hard-square model it is irreducible; (iv) the zeros of the resultant (equivalently the discriminant) of that characteristic polynomial — the algebraic locus of candidate singular points of the eigenvalues $\lambda(z)$ — have multiplicity two on the negative real axis for the computed widths, signalling eigenvalue crossings rather than singularities, in contrast to multiplicity one for hard squares; and (v) these observations disprove the 1989 zero-locus conjecture and show that Baxter's positive-axis result does not determine the analyticity of the partition function per site in the full complex $z$-plane. The authors propose a heuristic mechanism — extra eigenvalues becoming dominant in the necklace region — but supply no analytic construction, and the number of necklace regions (observed as $L_h/3-4$ for $L_h\le 27$) has unknown limiting behaviour.

Problem Statement

Determine the analytic structure, in the thermodynamic limit, of the hard-hexagon grand partition function per site throughout the complex fugacity plane, thereby completing Baxter's exact solution beyond the positive real axis. Concretely, as posed by Assis, Jacobsen, Jensen, Maillard and McCoy (2013):

1. (Necklace) Prove or disprove that the necklace of level-crossing regions in the left half of the complex $z$-plane persists as $L_h\to\infty$; determine whether in the thermodynamic limit the necklace is void of zeros, partially filled, or fully filled with partition-function zeros; and establish the limiting number and structure of its regions (observed to grow as $L_h/3-4$ for $L_h\le 27$, with the $L_h=30$ count equal to that of $L_h=24$).

2. (Eigenvalue branches) Identify which transfer-matrix eigenvalue branch(es) in addition to Baxter's $\kappa_{\pm}(z)$ become dominant in the necklace region as $L_h\to\infty$, and establish their algebraic and analytic origin — a concrete extension of Baxter's method tying the extra branches to the modular (Hauptmodul) structure of the model, as heuristically proposed in section 6.2 of the source.

3. (Algebraic regularities) Prove or refute that (a) the characteristic polynomial of the cylindrical-boundary hard-hexagon transfer matrix in the $P=0$ sector factorizes into two irreducible polynomials with integer coefficients for all widths $L_h$ (and decide whether factorization into more than two factors can occur), and (b) the zeros of the resultant (discriminant) of that characteristic polynomial on the negative real axis have multiplicity two for all $L_h$; and determine whether these properties are consequences of the model's Yang-Baxter integrability.

4. (Analytic continuation) Determine whether, and under what conditions, the partition function per site analytically continued from the positive real $z$-axis can be continued through the loci where the corresponding Baxter branch ceases to be the maximum-modulus transfer-matrix eigenvalue — that is, when does the free energy on the real axis determine analyticity throughout the complex plane — and whether this continuation property, exhibited by the integrable hard-hexagon model, extends to non-integrable models in the same universality class such as hard squares and the Ising model in a magnetic field.

These questions are jointly linked as one objective — the global complex-fugacity analytic structure of the hard-hexagon model. Parts (1)-(3) together are the required parts of that objective; part (4) is the source's framing of the same investigation's significance for generic (non-integrable) systems and is background context rather than a required part.

References

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