Rigorous derivation of the conjectured surface and corner free-energy product formulae for two-dimensional Potts, chromatic and loop models, and their unexplained boundary-exponent structures

For a large two-dimensional lattice model the logarithm of the partition function decomposes as $-MN f_b - M f_s - N f'_s - f_c$, and the corner free energy $f_c$ is the least controlled term. Vernier and Jacobsen (2012) combined Enting's finite lattice method with exact transfer-matrix enumeration to conjecture exact infinite product formulae, $\mathrm{e}^{f} = \prod_{k\ge 1}(1-q^k)^{\alpha_k}$ with $\alpha_k = \beta_k k + \gamma_k$ and $\beta,\gamma$ periodic, for the surface and corner free energies of the integrable $Q$-state Potts models on the square and triangular lattices, the chromatic polynomial, the Ising model and the fully-packed loop model, including under Jacobsen-Saleur boundary conditions. They left open: rigorous derivations of these products (in particular whether the corner transfer matrix can prove them), an explanation of the observed period doubling of boundary exponent sequences, completion of the $2\pi/3$-corner product for the triangular self-dual Potts case, resolution of a factor-of-three discrepancy between the lattice boundary-operator weight $h=1/48$ and the conformal-field-theory assignment $h=1/16$, a finite-lattice treatment of Ising fixed boundary conditions with macroscopic domain walls, analytic control of $q\to 1^-$ limits for products with linearly growing exponents, and a boundary-state interpretation of the corner results. Baxter (2017) subsequently proved the square-lattice Ising member by the free-fermion spinor method, and the surface formula for $Q>4$ is supported by a Bethe-ansatz derivation; all remaining members and all structural questions are open.

Background

For a classical spin model on an $M\times N$ lattice with open boundaries, the large-size expansion of the partition function reads $\beta^{-1}\log Z = -MN f_b - M f_s - N f'_s - f_c + O(\mathrm{e}^{-\gamma M}, \mathrm{e}^{-\gamma' N})$: $f_b$ is the bulk free energy per site, $f_s$ and $f'_s$ are surface free energies per unit edge length, and $f_c$ is the corner free energy, a geometry-dependent constant that is the hardest term to compute exactly. In the convention of Vernier and Jacobsen, $f \equiv \log Z$ is defined without the conventional minus sign and the decomposition reads $f_{M,N} = MN f_b + (M+N) f_s + f_c$; the product formulae quoted in this README are stated in that convention. At criticality $f_c$ diverges logarithmically in a way fixed by conformal field theory (CFT) through the Cardy-Peschel formula, which expresses the corner divergence in terms of the corner angle and the central charge (the number characterizing the CFT universality class).

Vernier and Jacobsen (2012) initiated the systematic exact study of $f_s$ and $f_c$ for integrable two-dimensional lattice models. Their tool is the finite lattice method (FLM), an inclusion-exclusion organization, introduced by Enting, that converts exact transfer-matrix enumerations of finite sublattices into infinite series for the thermodynamic free energies, together with a choice of expansion variable $q$ such that the model is trivial at $q=0$ and critical at $q\to 1^-$. The FLM cutoff $B(K)$ bounds the sublattice sizes retained and hence the order in $q$ attainable. For the self-dual $Q$-state Potts model on the square lattice (covering $Q \ge 4$), the antiferromagnetic (Berker-Kadanoff) critical curve of the square-lattice Potts model, the self-dual cubic curve of the triangular-lattice Potts model, the chromatic polynomial on the triangular lattice (the zero-temperature antiferromagnetic Potts partition function, which counts proper vertex colourings of the lattice), the Ising model at arbitrary temperature on both lattices, and the fully-packed loop model FPL$^2$ on the square lattice (a model whose configurations are two mutually excluding species of fully-packed loops — every vertex is visited by exactly one loop passage, black or red — carrying weights $n_1$ and $n_2$, integrable when $n_1 = n_2$), they found that the exponentiated free energies take the factorized form $\mathrm{e}^{f} = \prod_{k\ge 1}(1-q^k)^{\alpha_k}$ with $\alpha_k = \beta_k k + \gamma_k$ and $\beta, \gamma$ periodic functions of $k$ with periods between 4 and 16. Because the computed series cover several periods, this structure conjectures the products to all orders in $q$.

Two empirical regularities in these products remain unexplained. First, for Potts and loop models the exponent periods of the boundary products $\mathrm{e}^{f_s}$ and $\mathrm{e}^{f_c}$ are exactly twice that of the bulk product $\mathrm{e}^{f_b}$, while for Ising models all three periods coincide. Second, for products with linearly growing exponents ($\beta_k \ne 0$), the $q\to 1^-$ finiteness appears to be governed by the rule that $\prod_k (1-q^{ak-m})^{ak-m}/(1-q^{ak-n})^{ak-n}$ has a finite limit if and only if $m+n=a$; no analytic argument classifying such limits is known. The paper also generalized the FLM to particular boundary conditions, most prominently the Jacobsen-Saleur (JS) boundary conditions — a one-parameter family of conformally invariant boundary conditions for loop representations of the Potts model in which loops touching the boundary carry a modified weight $n_1$ parameterized by an integer $r$ — and conjectured product formulae for the boundary free-energy corrections at any $r$. In the $r\to\infty$ (Neumann) limit the corner divergence of the lattice computation corresponds to a boundary condition changing operator of conformal weight $h=1/48$, whereas the CFT prediction is $h=1/16$.

Baxter's corner transfer matrix (CTM), which builds the partition function as a product of four transfer matrices acting on quadrants and thereby makes corner physics directly accessible, is the natural candidate for proving such products, and Vernier and Jacobsen explicitly ask whether it can be made to do so. The natural benchmark is the free-fermion spinor method (Kaufman's reformulation of the Ising transfer matrix in terms of free fermions), which solved the Ising model: Baxter (2017) used it to prove the square-lattice Ising member of the conjecture family. For $Q \ne 2$ the Potts model has no free-fermion representation, the surface free energy of the self-dual model already required the full Bethe-ansatz machinery (the exact-solution technique for one-dimensional quantum and integrable two-dimensional lattice models), and no derivation of any $Q \ne 2$ corner product exists; the supporting evidence remains the FLM series and a handful of specializations.

Problem Statement

Resolve the open problems that Vernier and Jacobsen left on the exact product formulae for the surface and corner free energies of two-dimensional exactly solved lattice models. The common objective is to convert their series-supported conjectures into rigorous results and to explain the structural regularities of the product forms; the surviving core, after the resolution of the square-lattice Ising member by Baxter, is the following.

(1) Prove or refute the conjectured product formulae for $\mathrm{e}^{f_s}$ and $\mathrm{e}^{f_c}$ for the $Q \ne 2$ members of the family: the self-dual $Q$-state Potts model and the antiferromagnetic (Berker-Kadanoff) Potts curve on the square lattice, the self-dual curve of the triangular-lattice Potts model, the chromatic polynomial on the triangular lattice, the fully-packed loop model FPL$^2$ on the square lattice, and the product expansions for the surface and corner free-energy corrections of the self-dual square-lattice Potts model under Jacobsen-Saleur boundary conditions at any integer boundary parameter $r$, together with the still-conjectural triangular-lattice Ising corner product. Establish in particular whether the corner transfer matrix can produce and prove these product forms, as Vernier and Jacobsen proposed. (The square-lattice Ising case is already proved; the $Q>4$ self-dual surface formula is supported by a Bethe-ansatz derivation; the triangular-lattice Ising corner product remains a conjecture, the available series being consistent with it but too short to fix the exponents unambiguously.)

(2) Explain the period-doubling phenomenon: why, for all Potts and fully-packed loop models studied, the exponent sequences of $\mathrm{e}^{f_s}$ and $\mathrm{e}^{f_c}$ have exactly twice the periodicity of that of $\mathrm{e}^{f_b}$ (periods $4/8/8$ for the self-dual Potts model on the square and on the triangular lattice, $8/16/16$ for the antiferromagnetic Potts model on the square lattice, $6/12/12$ for the chromatic polynomial on the triangular lattice and $8/16/16$ for FPL$^2$ on the square lattice), while for the Ising models on both lattices all three periodicities coincide (the triangular-lattice Ising surface and corner periods being entered as '8 (?)' in the source, the series there being too short to fix the product unambiguously).

(3) Complete the product-form determination for the corners of angle $2\pi/3$ in the triangular-lattice self-dual Potts case, where the available series give only 22 of the 24 exponents required for one period and the completion was estimated to require finite-lattice enumerations up to cutoff $B(48)$.

(4) Determine the origin of the discrepancy for JS boundary conditions in the $r \to \infty$ limit, where the lattice corner divergence corresponds to a boundary-operator weight $h = 1/48$ (a corner divergence governed by a boundary condition changing operator of conformal weight $h$, the quantity fixing its critical exponent) while the conformal-field-theory assignment is $h = 1/16$; establish whether non-commuting limits (of $q \to 1^-$, $r \to \infty$ and the conformal limit) are responsible.

(5) Construct a finite-lattice (or equivalent localized-excitation) series formalism that correctly treats Ising models with fixed, non-free boundary conditions, in which domain walls extend macroscopically (such as the corner between $+$ and $-$ fixed sides with two free sides) and the present FLM decomposition fails.

(6) Establish analytic control of the $q \to 1^-$ limits of products $\prod_{k\ge 1}(1-q^k)^{\beta_k k + \gamma_k}$ with linearly growing exponents, including a proof of the observed finiteness rule that $\prod_k (1-q^{ak-m})^{ak-m}/(1-q^{ak-n})^{ak-n}$ has a finite limit if and only if $m+n=a$ (apart from the trivial case $m=n$).

(7) Interpret the lattice corner results in conformal-field-theory boundary-state language, generalizing the conformal boundary state for the rectangular geometry, including a microscopic understanding of when the finite-lattice additivity of corner free energies is compatible with the corner-corner interactions arising from modular dependence.

A complete answer converts the surviving conjectures into theorems or refutations and supplies the requested explanations; progress on individual items is meaningful but partial.

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