Exact surface and corner free energies of the anisotropic triangular-lattice Ising model: proving or refuting Baxter's conjectured product formulae

For the ferromagnetic Ising model on the triangular lattice with free boundaries, the exact bulk free energy has been known since the 1950s, but the boundary (surface and corner) free energies are only conjecturally known. R. J. Baxter (2020) computed 108-term low-temperature series for the partition function of arbitrary convex lattice polygons using Kaufman's spinor method and extrapolated them to explicit infinite-product and summation formulae for the surface and corner partition factors, valid in the anisotropic ordered regime; in the isotropic limit the 60-degree corner formula agrees with the 2012 conjecture of Vernier and Jacobsen. These formulae have never been derived or refuted. Two obstructions are identified in the source itself: the individual corner factors are not uniquely fixed by the factorization (a gauge freedom, since only eleven convex lattice shapes exist and they all satisfy the same corner-number constraint), and the inversion-relation technique that settled the square-lattice case is blocked by essential singularities of the corner products at the elliptic-parameter boundary $a=q$. The problem is to establish the exact surface and corner free energies of the anisotropic triangular-lattice Ising model rigorously — including a resolution of the corner normalization ambiguity — or to refute the conjectures by exact or certified independent computation. This would complete the exact boundary free-energy program for the Ising model on the two fundamental planar lattices, whose square-lattice member was proven in 2017.

Background

The Ising model assigns spins $\sigma=\pm1$ to the sites of a lattice, with each nearest-neighbour pair interacting with coupling $K>0$ (Boltzmann weight $z=e^{-2K}$ for the low-temperature variable); on the triangular lattice the three edge directions can carry different couplings $K_1,K_2,K_3$ (the anisotropic model). For a finite lattice region with free (open) boundary conditions, the partition function $Z$ decomposes asymptotically into contributions from the interior (bulk), the boundary edges (surface), and the boundary corners. Concretely, for a convex polygon drawn on the triangular lattice with edges along the three lattice directions, Baxter writes the reduced partition function as $\widehat{Z} \;=\; 2\,\kappa_b^{\,n_b}\,\prod_{i=1}^{3}\kappa_{s,i}^{\,n_{s,i}}\,\kappa_{c,i}^{\,n_{c,i}}\,\tilde{\kappa}_{c,i}^{\,\tilde{n}_{c,i}},$ expected to hold in the sense that the ratio of the two sides tends to one when the area and all edges become large; $n_b$ counts all sites, $n_{s,i}$ the boundary sites on edges of type $i$, $n_{c,i}$ the 60-degree corners of type $i$, and $\tilde{n}_{c,i}$ the 120-degree corners of type $i$. The partition factors define free energies per site, per boundary site, and per corner via $-\beta f=\log\kappa$ with $\beta=1/(k_BT)$. On the triangular lattice the boundary edges come in three types and the corners in two angles (60 and 120 degrees), each in three types.

The bulk free energy of the triangular-lattice Ising model has been known exactly since 1950. For the square lattice, the surface free energy was obtained by McCoy and Wu in 1967, but corner free energies long resisted exact treatment. Vernier and Jacobsen (2012) developed about twenty-five-term series expansions for the corner free energies of the Ising model, the Potts model (its $q$-state generalization) and fully packed loop models (lattice models whose configurations are non-crossing loops filling the lattice) on the square and triangular lattices, and conjectured exact closed forms, among them the 60-degree corner free energy of the isotropic triangular-lattice Ising model. For the square lattice these conjectures were then proven: Baxter (2017) used Kaufman's spinor method — the expression of the Ising transfer matrix through free-fermion (Clifford-algebra) operators, which reduces the partition function to the square root of the determinant of a matrix assembled from simple edge-by-edge factors — together with inversion relations (functional equations of the model under inverting a coupling variable, which can recursively determine free energies) to derive the exact bulk, surface and corner free energies of the anisotropic square-lattice model in the thermodynamic limit; independently, Hucht (2017) obtained the exact partition function of the open square-lattice rectangle via the dimer/Pfaffian method (mapping the Ising model to a dimer model, a covering of the lattice by nearest-neighbour pairs, whose partition function is evaluated as a Pfaffian, the antisymmetric analogue of a determinant), from which the corner free energy follows. Baxter also settled the self-dual Potts model (the Potts model at its self-dual point, where it maps to itself under duality) the same year. The triangular lattice, however, remained open.

Baxter (2020) attacked the anisotropic triangular model below its critical temperature $T_c$. Using the spinor method for arbitrary convex lattice polygons, he obtained series to 108 terms for the bulk, surface and corner free energies, observed repeating patterns of period 24 (in the isotropic low-temperature variable $p$, related to $z$ by an infinite product, with elliptic nome $q=p^6$ — the nome is the standard parameter of the elliptic functions entering the formulae), and extrapolated to conjecture exact results: for the isotropic model the products and exponential-sum formulae (4.1)-(4.10) of that paper, and for the fully anisotropic model the products (5.19)-(5.26) and equivalent summation formulae (5.28)-(5.34), expressed through elliptic parameters $a_1,a_2,a_3$ with $a_1a_2a_3=q$ and couplings $z_j=a_j^{1/2}G(a_j,q)$ for an explicit infinite product $G$. The conjectures pass strong internal checks: the bulk factor reproduces the known exact bulk free energy; the isotropic 60-degree corner formula agrees with the Vernier-Jacobsen conjecture; setting $K_3=0$ recovers the proven square-lattice results; and the critical behaviour exhibits the logarithmic singularities (corner exponent $\alpha=0$) predicted by the conformal-invariance analysis of Cardy and Peschel (1988) for the corner free energy at criticality.

Nevertheless, all of this is extrapolation, not proof, and Baxter's paper identifies specific obstructions. First, the algebraic derivation that succeeded for the square lattice was not carried over (the paper states the surface free energies could presumably be derived by generalizing those methods, but this was not done). Second, the corner factors are not uniquely defined by the factorization: only eleven combinatorially distinct convex lattice shapes exist, and they all satisfy $n_1-\tilde{n}_1=n_2-\tilde{n}_2=n_3-\tilde{n}_3$, so the transformation $\kappa_{c,i}\to\rho_i\kappa_{c,i}$, $\tilde{\kappa}_{c,i}\to\tilde{\kappa}_{c,i}/\rho_i$ with $\rho_1\rho_2\rho_3=1$ leaves the conjectured factorization invariant — a gauge freedom that any derivation must resolve. Third, the inversion-relation technique is only partially effective here: the available relations determine the bulk factor and give one relation for the surfaces, which is insufficient; the naive corner relations are meaningless because the corner product $\kappa_c$ has an essential singularity at the elliptic-parameter boundary $a=q$ (so it cannot be analytically continued to smaller $a$), and $\tilde{\kappa}_c$ likewise cannot be continued past $a=q^{-1}$, blocking the continuation across inversion points that the square-lattice argument required. No later work has derived or refuted these formulae.

Problem Statement

Establish rigorously — prove or refute — the conjectured exact boundary free energies of the anisotropic triangular-lattice Ising model in the ferromagnetically ordered phase, and resolve the corner normalization ambiguity that the conjecture carries.

Setting: the ferromagnetic Ising model on the triangular lattice with couplings $K_1,K_2,K_3>0$ and low-temperature variables $z_i=e^{-2K_i}$, restricted to the ordered phase $T<T_c$, i.e. the interior of the triangular-lattice critical surface $z_1z_2+z_2z_3+z_3z_1<1$, with free boundary conditions, on convex polygons whose edges follow the three lattice directions. Writing $n_b$ for the number of sites, $n_{s,i}$ for boundary sites on edges of type $i$, $n_{c,i}$ for 60-degree corners of type $i$ and $\tilde{n}_{c,i}$ for 120-degree corners of type $i$, the conjecture (Baxter 2020, eq. (5.1)) is that, in the limit where the area and every boundary edge grow large, $\widehat{Z} \;=\; 2\,\kappa_b^{\,n_b}\,\prod_{i=1}^{3}\kappa_{s,i}^{\,n_{s,i}}\,\kappa_{c,i}^{\,n_{c,i}}\,\tilde{\kappa}_{c,i}^{\,\tilde{n}_{c,i}},$ with shape-independent factors: the bulk factor $\kappa_b$ exactly known, and the surface factors $\kappa_{s,i}$ and corner factors $\kappa_{c,i},\tilde{\kappa}_{c,i}$ given by the conjectured infinite products (5.19)-(5.26) of Baxter (2020), equivalently their exponential-sum forms (5.28)-(5.34), in the elliptic parametrization $a_1a_2a_3=q$, $z_j=a_j^{1/2}G(a_j,q)$; the isotropic specialization (4.1)-(4.10) with $q=p^6$ contains, at the 60-degree corner, the Vernier-Jacobsen (2012) conjecture. The conjectures are verified by series to order $p^{108}$ on all the convex shapes calculated and pass the known limiting checks (exact bulk, $K_3\to0$ square-lattice limit, Cardy-Peschel critical behaviour), but no derivation exists.

The problem has two coupled components:

(1) A mathematically rigorous determination of the surface factors $\kappa_{s,i}$ and the corner factors $\kappa_{c,i},\tilde{\kappa}_{c,i}$ throughout the ordered regime — for instance by deriving them from the finite-lattice partition function via an extension of the Kaufman spinor method or of the Pfaffian/dimer technique that settled the square-lattice case — either confirming the conjectured product and summation formulae or establishing a precisely stated corrected form; alternatively, a refutation of the conjectures as stated, by an exact or certified-precision independent computation of one of these boundary free energies that disagrees with the conjectured closed form.

(2) A resolution of the corner gauge freedom: since all eleven distinct convex lattice shapes satisfy $n_1-\tilde{n}_1=n_2-\tilde{n}_2=n_3-\tilde{n}_3$, the factorization alone fixes only gauge-invariant corner combinations and is invariant under $\kappa_{c,i}\to\rho_i\kappa_{c,i}$, $\tilde{\kappa}_{c,i}\to\tilde{\kappa}_{c,i}/\rho_i$ with $\rho_1\rho_2\rho_3=1$. A complete answer must either derive the corner factors in a canonical normalization, identifying the convention implicit in the conjectured products, or prove that only the gauge-invariant corner content is determinate and re-express the conjecture accordingly. A refutation under component (1) discharges this requirement — there are then no conjectured corner products to normalize — so component (2) applies only on the confirm-or-correct route.

A full solution completes the exact boundary free energies of the Ising model on the two fundamental planar lattices, settling at once the triangular-lattice corner conjectures of Vernier and Jacobsen (2012) and Baxter (2020).

References

1. R. J. Baxter, 'The bulk, surface and corner free energies of the anisotropic triangular Ising model', Proc. R. Soc. A 476, 20190713 (2020), DOI: 10.1098/rspa.2019.0713, https://doi.org/10.1098/rspa.2019.0713 (preprint arXiv:1909.00518 with subtitle 'series expansions and critical behaviour', https://arxiv.org/abs/1909.00518)
2. E. Vernier and J. L. Jacobsen, 'Corner free energies and boundary effects for Ising, Potts and fully packed loop models on the square and triangular lattices', J. Phys. A: Math. Theor. 45, 045003 (2012), DOI: 10.1088/1751-8113/45/4/045003, https://doi.org/10.1088/1751-8113/45/4/045003 (preprint arXiv:1110.2158, https://arxiv.org/abs/1110.2158)
3. R. J. Baxter, 'The bulk, surface and corner free energies of the square lattice Ising model', J. Phys. A: Math. Theor. 50, 014001 (2017), DOI: 10.1088/1751-8113/50/1/014001, https://doi.org/10.1088/1751-8113/50/1/014001 (preprint arXiv:1606.02029, https://arxiv.org/abs/1606.02029)
4. A. Hucht, 'The square lattice Ising model on the rectangle I: finite systems', J. Phys. A: Math. Theor. 50, 065201 (2017), DOI: 10.1088/1751-8121/aa5535, https://doi.org/10.1088/1751-8121/aa5535 (preprint arXiv:1609.01963, https://arxiv.org/abs/1609.01963)
5. A. Hucht, 'The square lattice Ising model on the rectangle II: finite-size scaling limit', J. Phys. A: Math. Theor. 50, 265205 (2017), DOI: 10.1088/1751-8121/aa6b7a, https://doi.org/10.1088/1751-8121/aa6b7a
6. R. J. Baxter, 'Surface and corner free energies of the self-dual Potts model', arXiv:1606.01616 (2016, revised 2017), https://arxiv.org/abs/1606.01616
7. B. Kaufman, 'Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis', Phys. Rev. 76, 1232-1243 (1949), DOI: 10.1103/PhysRev.76.1232, https://doi.org/10.1103/PhysRev.76.1232
8. J. L. Cardy and I. Peschel, 'Finite-size dependence of the free energy in two-dimensional critical systems', Nucl. Phys. B 300, 377-392 (1988), DOI: 10.1016/0550-3213(88)90604-9, https://doi.org/10.1016/0550-3213(88)90604-9