Rigorous derivation of the surface and corner free energies of the anisotropic triangular-lattice Ising model: Baxter's product-form conjectures and the corner-factor gauge ambiguity

For the ferromagnetic Ising model on the triangular lattice with three unequal couplings in the ordered phase, Baxter (2019/2020) computed low-temperature series for the partition function of arbitrary convex lattice polygons — 108 terms for the bulk, surface and corner free energies — and, by extrapolating observed period-24 patterns, conjectured exact product and summation formulae for the surface and corner partition factors in elliptic-function variables. He derived nothing algebraically: the individual corner factors remain undetermined up to a gauge freedom $\kappa_{c,i}\to\rho_i\kappa_{c,i}$ , $\tilde\kappa_{c,i}\to\tilde\kappa_{c,i}/\rho_i$ with $\rho_1\rho_2\rho_3=1$ that the factorization over the eleven convex lattice polygons cannot fix, and the inversion-relation method that proved the square-lattice analogues is blocked by insufficient functional relations and by essential singularities of the corner factors at the elliptic boundaries $a=q$ and $a=q^{-1}$ . The problem is to establish these boundary free energies rigorously: prove, correct, or refute the conjectured formulae, and clarify the analytic structure that governs the inversion-relation route.

Background

The Ising model places spins $\sigma=\pm1$ on lattice sites with ferromagnetic nearest-neighbour couplings, of reduced energy $-K\sigma\sigma'$ per bond (measured in units of $k_BT$ , so that $K$ already includes the inverse temperature $\beta=1/(k_BT)$ ); reversing one spin of a bond then costs the relative Boltzmann weight $z=\mathrm{e}^{-2K}$ . The triangular lattice has three edge directions, so a fully anisotropic model has three couplings $K_1,K_2,K_3>0$ and weights $z_j=\mathrm{e}^{-2K_j}$ , $j=1,2,3$ . In the ordered (low-temperature, $T<T_c$ ) phase, the reduced partition function $\widehat{Z}$ of a large finite region (the partition function with the leading all-aligned ground-state edge weights divided out) decomposes asymptotically into contributions of distinct geometric origin: a bulk term proportional to the number $n_b$ of sites, surface terms proportional to the numbers $n_{s,i}$ of boundary sites of each of the three surface types $i$ , and corner terms of order unity. Convex polygons drawn on the triangular lattice have corners of angles $60^\circ$ and $120^\circ$ only; a surface lying along edges of type $i$ is said to be of type $i$ , and a $60^\circ$ corner at which surfaces of types $i-1$ and $i+1$ (mod 3) meet is itself said to be of type $i$ , and likewise for $120^\circ$ corners. Baxter writes the factorization 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}},

where $\kappa_b$ , $\kappa_{s,i}$ , $\kappa_{c,i}$ , $\tilde\kappa_{c,i}$ are the partition factors per site, per surface site of type $i$ , per $60^\circ$ corner of type $i$ and per $120^\circ$ corner of type $i$ (free energies via $-\beta f=\log\kappa$ ), in the limit where every surface is large. Corner free energies also have a universal aspect: at criticality, conformal invariance fixes the logarithmic corner singularity in terms of the central charge $c$ (the number labelling the universality class of a critical two-dimensional system) — the Cardy–Peschel predictions.

The problem sits in the exactly-solved tradition of the two-dimensional Ising model. Onsager (1944) obtained the square-lattice bulk free energy; Kaufman (1949) reformulated the solution with her spinor method (a fermionic representation in which the row transfer matrix is diagonalized by spinor, i.e. Clifford-algebra, methods); McCoy and Wu (1967) obtained the square-lattice surface free energy. For the triangular lattice the bulk free energy has been known since 1950 (Houtappel; Stephenson's 1964 derivation). Vernier and Jacobsen (2012) combined a finite-lattice method with exact transfer-matrix enumerations to obtain series of typically twenty-five terms for the bulk, surface and corner free energies of Potts-type models on the square and triangular lattices, conjecturing closed product forms where their series sufficed; for the triangular Ising model they could not obtain enough terms to conjecture the surface and the $120^\circ$ corner free energies. The square-lattice conjectures were then proved: Baxter (2017) gave an algebraic derivation using the spinor method together with inversion relations, and Hucht (2017) an independent derivation evaluating the finite-rectangle partition function as a Pfaffian (the signed sum whose square is the determinant of an antisymmetric matrix, the standard tool for free-fermion/dimer evaluations) and taking the thermodynamic limit; Hucht's determinant machinery was extended in 2021. Baxter also derived the surface and corner free energies of the self-dual Potts model (2016). An inversion relation is a functional equation for a free-energy factor obtained by inverting the transfer matrix under maps such as $z_j\to1/z_j$ ; two independent such relations, together with analyticity in the appropriate domain, suffice to determine the factor — the classical technique of Baxter (1982).

Baxter's 2019 paper (published 2020) brought this program to the anisotropic triangular lattice. Using Kaufman's spinor method he calculated low-temperature series for $Z$ of arbitrary convex triangular-lattice polygons, obtaining 108 terms for the free energies, observed repeat patterns of period 24 in the exponents of the product expansions, and extrapolated to conjecture exact product and summation formulae for $\kappa_{s,i}$ , $\kappa_{c,i}$ , $\tilde\kappa_{c,i}$ in elliptic-function variables: parameters $a_1,a_2,a_3$ with $a_1a_2a_3=q$ and nome $q$ (the elliptic-function expansion parameter), physical regime $q<a_i\le1$ . His checks: the bulk factor agrees with the classical exact result; the isotropic $60^\circ$ corner agrees with Vernier and Jacobsen's conjecture; setting $K_3=0$ (hence $a_3=1$ ) regains the proven square-lattice formulae; the critical behaviour shows the Cardy–Peschel logarithmic corner singularities. However, unlike the square-lattice case, nothing is derived algebraically for the boundary factors — they are conjectures. Baxter identifies two structural obstructions. First, the corner factors are not uniquely fixed: the partition function involves them only through the combinations $\kappa_{c,i}\tilde\kappa_{c,i}$ and the two triple products, because the eleven distinct convex shapes available all satisfy $n_{c,1}-\tilde n_{c,1}=n_{c,2}-\tilde n_{c,2}=n_{c,3}-\tilde n_{c,3}$ ; one is free to make the gauge transformation $\kappa_{c,i}\to\rho_i\kappa_{c,i}$ , $\tilde\kappa_{c,i}\to\tilde\kappa_{c,i}/\rho_i$ with $\rho_1\rho_2\rho_3=1$ . Second, the inversion-relation technique that closed the square-lattice argument breaks down: the available relations determine the bulk factor and give only one (insufficient) relation for the surfaces, and the naive corner relations are meaningless because the corner factors are functions of a single elliptic parameter $a$ : the $60^\circ$ factor $\kappa_c(a,q)$ (with $\kappa_{c,i}=\kappa_c(a_i,q)$ ) has an essential singularity at $a=q$ , and the $120^\circ$ factor $\tilde\kappa_c(a_i\mid a_{i+1},a_{i-1}\mid q)$ cannot be analytically continued beyond $a=q^{-1}$ ; these boundaries $a=q$ and $a=q^{-1}$ lie at and beyond the edges of the physical regime $q<a_i\le1$ , blocking the continuation across inversion points that the method requires.

Problem Statement

Determine, with mathematical rigor, the exact surface and corner free energies of the anisotropic triangular-lattice Ising model in the ordered phase $T<T_c$ . Specifically: prove, correct, or refute Baxter's conjectured product and summation formulae — eqs. (5.19)–(5.20) for the surface factors $\kappa_{s,i}$ , eqs. (5.21)–(5.22) for the $60^\circ$ corner factors $\kappa_{c,i}$ , and eqs. (5.23)–(5.26) for the $120^\circ$ corner factors $\tilde\kappa_{c,i}$ of Proc. R. Soc. A 476, 20190713 (2020) — which are currently supported only by 108-term low-temperature series, limit agreements, and critical-behaviour checks. For the two corner groups the target is Baxter's individual factors in the normalisation he states (eqs. (5.21)–(5.22) and (5.23)–(5.26), with $\kappa_{c,i}=\kappa_c(a_i,q)$ ); because the partition function fixes these factors only up to the gauge freedom $\kappa_{c,i}\to\rho_i\kappa_{c,i}$ , $\tilde\kappa_{c,i}\to\tilde\kappa_{c,i}/\rho_i$ with $\rho_1\rho_2\rho_3=1$ , an answer must be consistent with the gauge-invariant content (the products $\kappa_{c,i}\tilde\kappa_{c,i}$ and the two triple products), and a pure regauge that leaves this content unchanged counts as neither a proof, a correction, nor a refutation. A complete answer must: (i) settle each of the three groups of formulae independently, rigorously rather than by series extrapolation — by proving the conjectured formula, by proving a corrected formula, or by refuting it; because the groups are independent, a refutation of one group settles only that group and does not by itself resolve the task, so a complete answer must settle all three groups; and (ii) clarify the analytic structure of the corner factors at the elliptic boundaries $a=q$ and $a=q^{-1}$ (the essential singularities that currently block analytic continuation of the inversion-type relations), either by resolving that continuation or by establishing the boundary factors through a route for which the obstruction is bypassed, the latter discharging (ii). Whatever the route, the result must reproduce the known series coefficients, the isotropic-triangular limit, the $K_3\to0$ square-lattice limit, and the Cardy–Peschel critical behaviour.

References

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