Thermodynamic limit of the elliptic SOS model with domain-wall boundaries at general anisotropy, and its connection to the three-colouring functional equations

The elliptic solid-on-solid model (8VSOS) with domain-wall boundary conditions has an exactly known finite-size partition function: Galleas (Nucl. Phys. B 867 (2013) 855) derived a linear functional equation for it from the dynamical Yang–Baxter algebra and solved it as a finite sum over permutations and as a multiple contour integral, valid for arbitrary anisotropy. The paper explicitly left open the model's bulk thermodynamics — the boundary-dependent free energy in the thermodynamic limit at general anisotropy — and the relation between its functional equation and the independent functional equations of Razumov and Stroganov for the three-colouring specialization at anisotropy $2\pi i/3$, where Rosengren had conjectured an explicit free energy in terms of special polynomials. The third direction left open by the paper — proving the conjectured differential equation and Painlevé-type recurrence for those polynomials — has since been resolved independently by Rosengren's 2013–2015 series, which proves a non-stationary Schrödinger equation for the polynomials and identifies their specializations with Painlevé VI tau functions. The surviving open problem is therefore the thermodynamic limit of the 8VSOS domain-wall partition function at general anisotropy, including a rigorous derivation of the conjectured three-colouring free energy, together with the structural connection between the two families of functional equations; both components remain open.

Background

The eight-vertex solid-on-solid model (8VSOS, also called the elliptic SOS model) is an integrable two-dimensional lattice model of statistical mechanics. A solid-on-solid model assigns an integer-valued height to every site of a square lattice, with adjacent heights constrained to differ by $\pm\gamma$, where $\gamma$ is the anisotropy parameter; each elementary square of the lattice then carries a Boltzmann weight depending on its four corner heights. In the elliptic regime the weights are built from Jacobi theta functions, and the model depends additionally on a continuous dynamical parameter (denoted $\tau$ or $\theta$) that enters through the heights $h=\tau+n\gamma$, and on the elliptic nome $p$ (the parameter controlling the double periodicity of the elliptic functions). Integrability is governed by the dynamical Yang–Baxter equation, the elliptic analogue of the quantum Yang–Baxter equation in which the $R$-matrix (the operator collecting the model's elementary Boltzmann weights) depends on both a spectral parameter and this extra dynamical variable, which shifts under the algebra's exchange relations. The eight-vertex model solved by Baxter in 1972 is the periodic-boundary ancestor of this family. Domain-wall boundary conditions (DWBC), introduced by Korepin in 1982 for the six-vertex model, fix the boundary heights in a staircase profile instead of imposing periodicity. They are the boundary conditions under which lattice-model partition functions can be computed exactly without solving Bethe-ansatz-type equations (the coupled algebraic equations that otherwise must be solved to diagonalize the transfer matrices of integrable models): the six-vertex domain-wall partition function has a closed determinantal form, and Korepin and Zinn-Justin (2000) showed that the bulk free energy per site extracted from it in the thermodynamic limit differs from the free energy of the same model with periodic boundary conditions — boundary conditions that occupy zero fraction of the lattice nevertheless change the leading asymptotics. Computing such boundary-dependent bulk free energies is therefore a genuine analytic problem, distinct from the classical periodic solution. For the elliptic SOS model with DWBC, several exact finite-size representations of the partition function $Z_\theta$ are known: the Izergin–Korepin-type determinant of Rosengren (2009), the elliptic-weight-function representation of Pakuliak, Rubtsov and Silantyev (2008), and the functional-equation approach of Galleas (2013). Galleas derived from the dynamical Yang–Baxter algebra a linear functional equation relating $Z_{\theta-\gamma}$ at shifted dynamical argument to functions with one spectral parameter replaced by an auxiliary value, with explicit theta-function coefficients, and solved it as a finite sum over permutations and as a multiple contour integral, valid for arbitrary anisotropy $\gamma$, arbitrary dynamical parameter, and arbitrary lattice size. At the special anisotropy $\gamma=2\pi i/3$ the model degenerates to Baxter's three-colouring model (1970), whose configurations are proper 3-colourings of the square lattice. At this point three additional structures are known: Razumov and Stroganov (2009) derived an independent system of functional equations for the domain-wall partition function; Rosengren (2011) expressed the partition function through polynomials $p_n$ and two-variable polynomials, for which Mangazeev and Bazhanov (2010) conjectured a partial differential equation and a Painlevé-type recurrence (a recurrence of the kind obeyed by the Painlevé transcendents, the classical special functions defined by nonlinear ordinary differential equations); and Rosengren (2011) also conjectured, via a formal but non-rigorous limiting argument, an explicit formula for the free energy of the three-colouring model with domain-wall boundaries. For general anisotropy no thermodynamic-limit result is available, and the relation between the two functional-equation frameworks has, in the source paper's own words, 'eluded us so far'.

Problem Statement

Galleas's refined functional relations for the elliptic SOS model with domain-wall boundaries left three directions open; the third — proving the conjectured differential equation and recurrence relation for the associated polynomials — has since been resolved by Rosengren's independent series of papers, so the present problem is the surviving core consisting of the other two directions. (1) Thermodynamic limit at general anisotropy. Let $Z_\theta$ denote the partition function of the elliptic SOS model with domain-wall boundaries on an $L\times L$ lattice. Compute the bulk free energy $f=-\lim_{L\to\infty}L^{-2}\log Z_\theta$ at the homogeneous point (all spectral parameters equal, with the normalization used stated explicitly) for general anisotropy $\gamma$, general dynamical parameter, and general nome; the derivation must state which exact finite-size representation of $Z_\theta$, if any, it uses. A rigorous derivation of $f$ at general anisotropy is required, and the comparison with the periodic-boundary free energy of Baxter's eight-vertex solution — settling whether the two coincide or differ — together with the rigorous derivation (or refutation) of Rosengren's conjectured three-colouring free energy at the special value $\gamma=2\pi i/3$, must then be given as explicit corollaries; the three-colouring derivation in particular must be rigorous rather than purely formal. (2) Functional-equation connection at the three-colouring point. Establish the precise relation between Galleas's refined functional equation for $Z_\theta$ and the Razumov–Stroganov functional equations for the three-colouring domain-wall partition function at $\gamma=2\pi i/3$: whether the two systems are equivalent, whether one is a specialization of the other, or whether no such relation holds, with the elliptic-function identities and normalization conversions made explicit whenever a relation is exhibited. Both directions are required deliverables, and they concern the same object and were posed together in the source; the three-colouring point is the only anisotropy value where alternative functional equations and a free-energy conjecture exist, so that resolving direction (2) provides the natural structural input and cross-check for direction (1) at that point.

References

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