Rigorous derivation of the corner free energy product formulae for integrable Potts and loop models, and the periodicity structure of their exponents

Vernier and Jacobsen (2012) combined the finite-lattice method with exact transfer-matrix enumerations to conjecture exact product formulae, $\prod_{k\ge1}(1-q^k)^{\beta_k k+\gamma_k}$ with periodically repeating coefficient sequences, for the surface and corner free energies of several integrable two-dimensional lattice models: the self-dual and zero-temperature antiferromagnetic $Q$-state Potts models on the square and triangular lattices, the chromatic polynomial on the triangular lattice, the fully-packed loop model FPL$^2$, and the Ising models. They explicitly posed as open problems the rigorous first-principles derivation of these products — in particular whether the corner transfer matrix determines the corner free energy exactly — together with an explanation of the observed periodicity patterns (boundary-related exponents having twice the bulk period in Potts and loop models) and a conformal-field-theory interpretation of the lattice corner results. Later work by Baxter rigorously established the square-lattice Ising case, derived the surface free energy of the self-dual square-lattice Potts model by open-boundary Bethe ansatz in a preprint that remains unpublished, and strongly corroborated the triangular-lattice Ising formulae; the corner free energies of the Potts and loop models nevertheless remain conjectural. The open problem is to prove or refute the remaining product formulae from first principles, to explain the periodicity structure of their exponents, and to reconcile the lattice corner free energies with conformal boundary field theory, including the unexplained factor-of-three discrepancy in the boundary-operator weight of the $r\to\infty$ Jacobsen–Saleur limit.

Background

Consider a two-dimensional lattice model on an $M\times N$ rectangle with open boundaries. Away from criticality its free energy admits the expansion $\beta^{-1}\log Z=-MN f_b-M f_s-N f_s'-f_c+O(e^{-\gamma M},e^{-\gamma' N})$, where $f_b$ is the bulk free energy, $f_s$ and $f_s'$ the surface free energies of the two boundary directions, and $f_c$ the corner free energy. Cardy and Peschel showed that at criticality the corner free energy is universal: for a corner of opening angle $\Theta$ it diverges logarithmically as $f_c(\Theta)\sim \frac{c}{24}\left(\frac{\pi}{\Theta}-\frac{\Theta}{\pi}\right)\ln L$ (up to the overall sign convention for the free energy), where $c$ is the central charge of the conformal field theory (CFT) governing the continuum limit. In contrast to the bulk and surface free energies, which are non-universal and were known exactly for many solvable models, essentially nothing was known about $f_c$ from the side of exact solvability: its computation requires control of the full finite-size transfer-matrix spectrum, not just the leading eigenvalue.

Vernier and Jacobsen attacked this gap by combining the finite-lattice method (FLM) — exact enumeration of finite lattices up to a size cutoff $B(n)$, arranged so that boundary contributions cancel order by order — with exact transfer-matrix techniques, producing long series for $e^{f_b}$, $e^{f_s}$ and $e^{f_c}$ (in their sign convention the free energy is $\log Z$ itself) around the trivial fixed points at $|Q|\to\infty$ (the chromatic-polynomial expansions being about $Q\to-\infty$) or $1/T$. For every model studied the series could be resummed into products $\prod_{k=1}^{\infty}(1-q^k)^{\alpha_k}$, where the exponent has the form $\alpha_k=\beta_k k+\gamma_k$ with $\beta_k$ and $\gamma_k$ periodic functions of $k$ with short periods (4–24), which allowed the products to be conjectured to all orders. The models comprise: the self-dual $Q$-state Potts model on the square and triangular lattices (the Potts model assigns each site a spin with $Q$ values and weights nearest neighbours by $e^{K\delta_{\sigma_i,\sigma_j}}$; on the integrable self-dual curves the loop weight of its Fortuin–Kasteleyn cluster representation is $n=\sqrt{Q}$, and the expansion parameter is fixed by $n=q+q^{-1}$, where $q$ is the quantum-group deformation parameter of the $U_q(sl_2)$ symmetry); the zero-temperature antiferromagnetic Potts model on the square lattice, with $Q=-(q-q^{-1})^2$; the chromatic polynomial ($Q$-colouring problem) on the triangular lattice, whose bulk free energy was solved exactly long ago; the fully-packed loop model FPL$^2$ — two mutually excluding families of fully-packed loops on the square lattice, integrable when the two loop weights are equal, with $U_q(sl_4)$ symmetry and central charge $c=3$; and the Ising model ($Q=2$) on the square and triangular lattices, where the natural parameter $q$ is the elliptic nome built from the complete elliptic integrals rather than a quantum-group parameter. The limit $q\to 1^-$ is critical for all these models, and the asymptotic analysis of the products reproduces the Cardy–Peschel angular dependence and yields the correlation-length divergence. As a representative example, the conjectured corner free energy of the self-dual square-lattice Potts model reads

$e^{f_c}=\prod_{k\ge1}\frac{1}{(1-q^{8k-6})(1-q^{8k-4})^{4}(1-q^{8k-2})},\qquad \sqrt{Q}=q+q^{-1}.$

The corner transfer matrix (CTM) of Baxter is a natural candidate for proving such products: it represents the Boltzmann weights of an entire lattice quadrant, and its eigenvalues famously have product form in the elliptic nome. The source also studied special boundary conditions — the Jacobsen–Saleur (JS) boundary conditions, defined in the loop representation of the Potts model by a boundary loop weight parameterised by an integer $r$ — and compared the corner free energies with CFT predictions involving boundary-condition-changing operators (local operators inserted where the boundary condition changes, whose conformal weights control the corner divergence).

Subsequent literature has settled part of this programme: Baxter derived the square-lattice Ising free energies rigorously by the spinor method, and the surface free energy of the self-dual square-lattice Potts model by open-boundary Bethe ansatz (in a preprint that remains unpublished); the triangular-lattice Ising products are strongly corroborated by 108-term series but remain conjectural. What remains is the subject of the problem stated below.

Problem Statement

The problem left open by Vernier and Jacobsen (2012), narrowed by the rigorous progress since described, has three linked components serving one objective — to establish exactly, rigorously and structurally the corner free energies of the integrable two-dimensional lattice models:

1. (Derivation.) Derive from first principles, rigorously, the exact surface and corner free energies of the models for which the product formulae remain conjectural: the self-dual $Q$-state Potts model on the square lattice, whose conjectured corner formula reads $e^{f_c}=\prod_{k\ge1}\frac{1}{(1-q^{8k-6})(1-q^{8k-4})^{4}(1-q^{8k-2})}$ with $\sqrt{Q}=q+q^{-1}$ in the source's sign convention; together with the self-dual Potts model on the triangular lattice, the zero-temperature antiferromagnetic Potts model on the square lattice, the chromatic polynomial on the triangular lattice, the fully-packed loop model FPL$^2$, and the triangular-lattice Ising surface and corner products, including the $2\pi/3$-corner product conjectured by Baxter (2020), which goes beyond the products explicitly conjectured in the 2012 source (the square-lattice Ising case and the self-dual Potts surface free energy having already been derived). A solution proves or refutes these product formulae, including the all-orders periodicity of their exponent sequences; the whole set of models listed is required on the same footing, and a rigorous result for any single model counts as partial progress only. The source's related question — whether the corner transfer matrix determines the corner free energy exactly, which if possible 'one could hope to prove our formulae for $e^{f_c}$' — remains open but is not itself a required deliverable: a derivation need not proceed via the corner transfer matrix.

2. (Structure.) Explain the observed periodicity patterns of the exponents: why the periods of the boundary-related quantities $e^{f_s}$ and $e^{f_c}$ are twice that of $e^{f_b}$ in all Potts and loop models (where $q$ is the quantum-group deformation parameter), why the three periods coincide for the Ising models (where $q$ is the elliptic nome), and why the periods are unchanged when the same model is solved on different lattices — questions the source explicitly connects to the corner transfer matrix and to conformally invariant boundary states.

3. (CFT reconciliation.) Interpret the lattice corner free energies in conformal boundary field theory: give a boundary-state account of the corner contributions, including the compatibility of the additivity of corners built into the finite-lattice method with the corner–corner interactions that arise in CFT from the dependence of rectangle partition functions on the modular parameter, and the origin of the discrepancy between the lattice-inferred boundary-operator weight $h=1/48$ and the CFT assignment $h=1/16$ in the $r\to\infty$ limit of the Jacobsen–Saleur boundary conditions.

These components are jointly the source's own emphasis (its Conclusion and §§3.2.8, 4.2 and 4.7): a resolution establishes the exactness — or failure — of the corner free energy products and simultaneously accounts for their algebraic structure and their conformal interpretation.

References

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