Rigorous derivation of the conjectured surface and corner free energies of the anisotropic triangular-lattice Ising model, and resolution of the corner-factor gauge ambiguity

For the anisotropic Ising model on the triangular lattice with free boundaries in the ferromagnetically ordered phase, Baxter (2020) computed 108-term low-temperature series for the bulk, surface, and corner contributions to the partition function of arbitrary convex polygonal domains and, by extrapolating observed repeat patterns, conjectured exact product (and equivalent summation) forms for the surface partition factor and for the two types of corner partition factors, expressed in elliptic function variables. The bulk factor is known exactly, the square-lattice analogues are proven, and several limits of the conjectures check out, but for the triangular lattice the boundary factors remain conjectures with no derivation. Moreover, the corner factors are only defined up to a gauge freedom (multiplication by arbitrary functions $\rho_i$ with $\rho_1\rho_2\rho_3=1$), the inversion-relation technique that succeeded on the square lattice provably fails here, and the conjectured corner products have essential singularities that block analytic continuation across the inversion points. The open problem is to prove or refute the conjectured exact boundary free energies, to fix the corner normalization canonically, and to settle the analytic structure that presently obstructs the inversion method. No resolution is known in the literature to date.

Background

The Ising model places spins $\sigma=\pm 1$ on the sites of a lattice with interaction energy $-K_j\sigma\sigma'$ along each of the three edge directions $j=1,2,3$ of the triangular lattice; when the couplings $K_j$ differ the model is anisotropic, and for sufficiently low temperature $T<T_c$ it is ferromagnetically ordered. Baxter studied finite convex polygonal domains of this lattice with free (open) boundaries, using Kaufman's spinor method — the fermionic diagonalization of the Ising transfer matrix introduced by Onsager and Kaufman — to expand the partition function $Z$ at low temperature to high order. Writing $z_j=e^{-2K_j}$ for the Boltzmann weight of an unsatisfied bond, the expansions are organized in an elliptic variable $p$ (defined by $z=p\prod_{k\ge 1}(1-p^{24k-20})(1-p^{24k-4})/[(1-p^{24k-16})(1-p^{24k-8})]$), and, in the anisotropic case, in three elliptic parameters $a_1,a_2,a_3$ with $a_1a_2a_3=q$ (with $q$ the elliptic nome, $q=p^6$ in the isotropic case), related to the $z_j$ through a standard elliptic product $z_j=a_j^{1/2}G(a_j,q)$.

The central structural observation is an asymptotic factorization: for a large convex domain with $n_b$ sites in total (the paper's bookkeeping counts every site, boundary and corner sites included), $n_{s,i}$ boundary sites lying along edges of type $i$ (counting each corner site as belonging to both its adjacent surfaces), $n_{c,i}$ corners of opening angle $60^\circ$ of type $i$, and $\tilde n_{c,i}$ corners of opening angle $120^\circ$ of type $i$, the partition function behaves as

$$\widehat Z = \kappa_b^{\,n_b}\prod_{i=1}^{3}(\kappa_{s,i})^{n_{s,i}}(\kappa_{c,i})^{n_{c,i}}(\tilde\kappa_{c,i})^{\tilde n_{c,i}},$$

with shape-independent factors $\kappa_b$, $\kappa_{s,i}$, $\kappa_{c,i}$, $\tilde\kappa_{c,i}$ — the partition functions per site, per boundary edge-site, per $60^\circ$ corner, and per $120^\circ$ corner. Here $\widehat Z$ is the partition function with the two fully ordered ground states and their bond energy divided out, $\widehat Z = Z/(2\,\mathrm{e}^{\sum_{\text{bonds}} K_j})$, the sum running over all bonds of the domain, so that $\widehat Z \to 1$ as $T \to 0$ (this overall normalization convention — the treatment of the two ordered ground states — varies between equations of the source, and the operative conjectured content is the explicit infinite product forms for the $\kappa$ factors given below, each of which tends to $1$ as $T \to 0$). The corresponding free energies are $-\beta f=\log\kappa$ with $\beta=1/(k_BT)$. The bulk factor $\kappa_b$ has been known exactly since the 1950s and agrees with the series. For the boundary factors, Baxter's 108-term series displayed repeat patterns of period 24 (exponents linear in the product index $k$, and quadratic in $k$ for parts of the $120^\circ$ corner factor), permitting extrapolation to all orders and hence conjectured closed forms: $\kappa_{s,i}=\boldsymbol\kappa_s(a_i\,|\,a_{i+1},a_{i-1}\,|\,q)$, $\kappa_{c,i}=\boldsymbol\kappa_c(a_i,q)$, and $\tilde\kappa_{c,i}=\boldsymbol{\tilde\kappa}_c(a_i\,|\,a_{i+1},a_{i-1}\,|\,q)$, each an explicit infinite product over $k$ of factors $(1-q^{\alpha k+\beta}a^{\gamma})$ raised to explicit linear or quadratic powers, together with equivalent summation forms $\log\kappa=\sum_m F(p^m)/m$.

These conjectures are supported by nontrivial checks: the isotropic $60^\circ$ corner product reproduces the earlier conjecture of Vernier and Jacobsen (who had extracted isotropic corner free energies from transfer-matrix numerics); the singular parts near criticality agree with the conformal-invariance prediction of Cardy and Peschel that a corner of opening angle $\theta$ contributes a term proportional to $(\theta/\pi-\pi/\theta)(c/24)\log(\text{scale})$ with central charge $c=1/2$; and setting the third coupling $K_3=0$ degenerates the lattice to the square lattice, where the boundary factors have been proven exactly — for the square lattice, Baxter derived the bulk, surface and corner free energies rigorously in 2017 using the inversion relation method, a technique that determines a free energy by solving functional equations such as $\kappa(a)\kappa(q^2/a)=\text{known}$, combined with analyticity.

Three structural obstructions, however, keep the triangular-lattice boundary factors conjectural. First, no derivation is known: the square-lattice derivation does not carry over, because the required decomposition of $Z$ with a parameter-independent structure fails on the triangular lattice, and the transfer-matrix eigenvalue route (used for the square-lattice surface free energy by McCoy and Wu) has not been carried through for three edge directions. Second, a gauge non-uniqueness: only eleven distinct convex shapes exist on the triangular lattice, and they all satisfy $n_{c,i}-\tilde n_{c,i}$ equal for $i=1,2,3$, so the factorization only fixes the combined corner quantities $\kappa_{c,i}\tilde\kappa_{c,i}$ and the products $\kappa_{c,1}\kappa_{c,2}\kappa_{c,3}$, $\tilde\kappa_{c,1}\tilde\kappa_{c,2}\tilde\kappa_{c,3}$ — one may transform $\kappa_{c,i}\to\rho_i\kappa_{c,i}$, $\tilde\kappa_{c,i}\to\tilde\kappa_{c,i}/\rho_i$ with $\rho_1\rho_2\rho_3=1$ (the $\rho_i$ being arbitrary functions of the elliptic parameters, not merely constants) without changing any observable partition function. The individual corner factors therefore need a separate microscopic or canonical definition. Third, the inversion relations that do hold — two for the bulk (sufficient to determine it) and one for the surface factor (insufficient) — cannot be extended to the corner factors: the formal corner identities have no common convergence domain, because $\boldsymbol\kappa_c$ has an essential singularity at the elliptic boundary $a=q$ (and $\boldsymbol{\tilde\kappa}_c$ cannot be continued past $a=q^{-1}$), blocking the analytic continuation on which the inversion method relies.

Problem Statement

Establish rigorously the boundary free energies of the anisotropic triangular-lattice Ising model in the ferromagnetically ordered phase ($T<T_c$) with free boundaries — the problem left open by Baxter (2020) — by proving or refuting the conjectured exact forms. Specifically, for the factorization $\widehat Z=\kappa_b^{\,n_b}\prod_i(\kappa_{s,i})^{n_{s,i}}(\kappa_{c,i})^{n_{c,i}}(\tilde\kappa_{c,i})^{\tilde n_{c,i}}$ of large convex domains, where $\kappa_b$ is the exactly known bulk factor:

(1) Prove (or disprove) that the limiting surface factor $\kappa_{s,i}$ and the corner factors $\kappa_{c,i}$, $\tilde\kappa_{c,i}$ (taken in the canonical normalization fixed in (2)) are given by the conjectured product forms in the elliptic variables $a_1,a_2,a_3,q$ ($a_1a_2a_3=q$, $q<a_j\le 1$; equivalently the summation forms $\log\kappa=\sum_m F(p^m)/m$), and establish the shape-independent asymptotic factorization itself — for example by a transfer-matrix eigenvalue analysis generalizing the McCoy-Wu treatment of the square-lattice surface free energy, by Pfaffian/determinant representations with asymptotic (Toeplitz/Hankel-type) analysis in the spirit of the square-lattice rectangle solution, or by any other rigorous route.

(2) Resolve the corner normalization: since convex-shape factorization fixes only the combined corner quantities, give a canonical, microscopic definition of the individual $60^\circ$ and $120^\circ$ corner factors — that is, break the gauge freedom $\kappa_{c,i}\to\rho_i\kappa_{c,i}$, $\tilde\kappa_{c,i}\to\tilde\kappa_{c,i}/\rho_i$ with $\rho_1\rho_2\rho_3=1$, the $\rho_i$ being arbitrary functions of the elliptic parameters — and show that the conjectured products correspond to it (or determine the correct factors under it).

(3) As part of a complete solution, characterize the analytic structure of the corner factors at the elliptic boundary $a=q$ (and $a=q^{-1}$) — the essential singularities that currently make the formal corner inversion identities meaningless — and clarify whether and how analytic continuation across the inversion points, and hence an inversion-relation determination of the boundary factors, is possible.

These three components jointly serve one objective: replacing numerically supported conjectures by derived exact results for the boundary thermodynamics of the model, on the same footing as the proven square-lattice case. A derivation must reproduce the known checks: the exactly known bulk factor, the $K_3=0$ reduction to the proven square-lattice boundary free energies, the isotropic $60^\circ$ corner agreement with the Vernier-Jacobsen conjecture, and the Cardy-Peschel critical singularities. A positive resolution requires all three components jointly — the product forms proved for the canonically normalized corner factors together with the shape-independent asymptotic factorization, the canonical normalization itself, and the characterization of the analytic structure — so that proving the surface product alone does not suffice; a component-(2) outcome in which the canonically defined corner factors differ from the conjectured products refutes those product forms and therefore resolves the problem in the negative, while still supplying the canonical definition asked for in (2). A disproof of any of the conjectured product forms (for instance through an exact boundary free energy that demonstrably disagrees with the corresponding product) equally resolves the problem in the negative.

References

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