Foundations of the inversion-relation free energy of thermalized IRF lattice models and the identity of the coset factor $\mathcal{G}$

Solvable interaction-round-a-face (IRF) lattice models can be built from the data of an arbitrary rational conformal field theory (RCFT) $\mathcal{O}$ and two of its primary fields via a Baxterization ansatz. Gepner (2021) conjectured that away from criticality these models obey 'thermalized' inversion relations, obtained by replacing the sine functions of the critical inversion relations with Jacobi theta functions, and solved them by Baxter's inversion-relation method, obtaining the exact free energy in the four standard regimes, the critical exponent $\alpha$, and the dimension of the thermal perturbing operator of the fixed-point theory. This determines the infrared fixed point only up to a coset $\mathcal{C}=\mathcal{G}/\mathcal{O}$ (regimes III, IV) or $\mathcal{C}=\mathcal{O}/\mathcal{G}$ (regimes I, II), with $\mathcal{G}$ an unidentified conformal field theory. The problem is to give this exact-solution machinery a rigorous foundation and to complete the fixed-point identification: prove, or refute by an explicit model of the class, (i) the thermalized inversion relations for arbitrary IRF models, (ii) the analyticity of the logarithm of the normalized transfer-matrix eigenvalue in the annulus containing $w=1$ and the regime's second inversion endpoint, together with the convergence condition $|\tilde{\zeta}_r|\leq\lambda$ and the symmetry $d_{-m}=d_m$ of the Laurent coefficients, and (iii) identify the auxiliary theory $\mathcal{G}$. All three components remain open: later work extends the conjectural framework and verifies the ansatz case by case but neither proves nor refutes any conjecture, and $\mathcal{G}$ remains unknown.

Background

Two-dimensional solvable lattice models are statistical systems whose Boltzmann weights satisfy the Yang–Baxter equation (YBE), a cubic relation on the R-matrix that guarantees commuting row-to-row transfer matrices and underlies exact solutions. In interaction-round-a-face (IRF) models, the fluctuating variables ('heights') live on the vertices of a square lattice, neighbouring heights obey a fusion-type admissibility condition, and a Boltzmann weight is attached to each elementary face. The classical exactly solved examples are the eight-vertex solid-on-solid model of Andrews, Baxter and Forrester (the ABF model) and the affine-Lie-algebra families of Jimbo, Miwa and Okado.

Gepner (1992) proposed building such models directly from conformal field theory. A rational conformal field theory (RCFT) is a two-dimensional conformal field theory whose chiral algebra has finitely many irreducible representations ('blocks'); its data include primary fields with conformal dimensions $\Delta_a$, fusion rules, and braiding matrices $B$ describing the monodromy of conformal blocks when two primaries are exchanged. From an RCFT $\mathcal{O}$ and two primary fields $h$, $v$ one defines a model IRF($\mathcal{O},h,v$): the heights are primaries of $\mathcal{O}$, admissibility is fusion with $h$, and the R-matrix — the 'Baxterization' — is a linear combination of the projectors $P^a$ onto the $n$ blocks of the fusion product $h\cdot h$, with coefficients built from $\sin(\zeta_j\pm u)$ of the spectral parameter $u$, where the crossing parameters $\zeta_a=\frac{\pi}{2}(\Delta_{a+1}-\Delta_a)$ and $\tilde{\zeta}_a=\frac{\pi}{2}(\tilde{\Delta}_{a+1}-\tilde{\Delta}_a)$ are fixed by the dimensions of the blocks of $h\cdot h$ and of $h\cdot\bar{h}$ respectively ($\bar{h}$ the conjugate field). This ansatz is conjectured to satisfy the YBE for any RCFT and is verified for the models based on Wess–Zumino–Witten theories (the RCFTs built from affine Lie algebras), such as the ABF and Jimbo–Miwa–Okado families.

Baxter's inversion-relation method extracts the free energy per site, $f=-k_BT\log\kappa$ with $\kappa$ the largest transfer-matrix eigenvalue, without solving the model. The eigenvalue obeys functional equations: a first inversion relation $R(u)R(-u)=\rho(u)\rho(-u)\mathbf{1}$, and a crossing relation acting as a second inversion relation, where $\rho$, $\tilde{\rho}$ are explicit products over the crossing parameters. Assuming the logarithm of the normalized eigenvalue is analytic in an annulus of the variable $w$ (with $\epsilon$ the temperature-like parameter, $w=\exp(-4\pi u/\epsilon)$ in regimes III and II and $w=\exp(2\pi u/\epsilon)$ in regimes IV and I) that contains the point $w=1$ together with the regime's second inversion endpoint, expanding in a convergent Laurent series and matching coefficients solves for the free energy. The method was introduced by Baxter and used in his solution of the hard-hexagon model.

Away from criticality the trigonometric (sine) solution must be replaced by an elliptic one: one 'thermalizes' by substituting the Jacobi theta function $\theta_1(u,q)$, with elliptic nome $q$ controlling the distance from criticality. Explicit elliptic Boltzmann weights are known only for special families (ABF-type restricted models and the affine-Lie-algebra families). For a general IRF($\mathcal{O},h,v$) the elliptic weights are unknown, so Gepner (2021) conjectures the thermalized inversion relations themselves, with $\rho$, $\tilde{\rho}$ the same products with $\sin$ replaced by $\theta_1$. Solving these by the inversion-relation method in the four standard regimes (ranges of $q$ and $u$ generalizing the usual regime classification of the solid-on-solid models) yields the free energy, subject to three unproven assumptions: analyticity of $\log\tilde{\kappa}(w)$ in the annulus containing $w=1$ and the regime's second inversion endpoint — the point $x=\exp(-4\pi\lambda/\epsilon)$ in regime III, and the point $w_0$ in the other regimes (in regime II, $w_0^2=\tilde{q}^{-1}x^2$ with $x=\exp(-4\pi\lambda/\epsilon)$ and $\tilde{q}=e^{-4\pi^2/\epsilon}$; in regimes IV and I, $x=-\exp(-\pi\lambda/\epsilon)$ and $\tilde{q}=-e^{-\pi^2/\epsilon}$, with $w_0=x^{-2}$ in regime IV and $w_0=-x^{-2}\tilde{q}$ in regime I) — symmetry of the Laurent coefficients $d_{-m}=d_m$, and the bound $|\tilde{\zeta}_r|\leq\lambda$ (with the crossing parameter $\lambda=\tilde{\zeta}_0=\frac{\pi}{2}\Delta_{\mathrm{adj}}$, $\Delta_{\mathrm{adj}}$ the adjoint-representation dimension), which is needed for convergence and is checked in many models but not proven. The solution reproduces Baxter's hard-hexagon free energy exactly.

The physical payoff is the infrared fixed point. The singularity of the free energy gives the critical exponent $\alpha$, from which the dimension of the thermal (energy-like) perturbing operator of the fixed-point theory follows: $\Delta_p=(1-\alpha)/(2-\alpha)=1-\Delta_{\mathrm{adj}}$ in regimes III and IV and $\Delta_p=\Delta_{\mathrm{adj}}$ in regimes I and II. This matches a coset structure: the fixed point is consistent with $\mathcal{C}=\mathcal{G}/\mathcal{O}$ in regimes III and IV and $\mathcal{C}=\mathcal{O}/\mathcal{G}$ in regimes I and II, where the coset (Goddard–Kent–Olive) construction builds a new conformal field theory from a pair of nested chiral algebras. The theory $\mathcal{G}$ is not determined by the computation — it is 'some unknown CFT'. In the known cases it can be read off: for the ABF model, which is IRF(SU(2)$_k$,[1],[1]), the regime III/IV fixed points are the unitary minimal models $\mathrm{SU}(2)_{k-1}\times\mathrm{SU}(2)_1/\mathrm{SU}(2)_k$ (Huse), and the Jimbo–Miwa–Okado families likewise match their established coset fixed points; but for a general RCFT $\mathcal{O}$ no identification of $\mathcal{G}$ exists.

Thus the exact-solution chain — inversion relations, free energy, critical exponent, fixed point — rests entirely on conjectures, and the fixed point is identified only up to the unknown factor theory $\mathcal{G}$. A further caveat: even the underlying trigonometric Baxterization ansatz is conjecturally, not provably, integrable for a general RCFT, so any general proof must engage with the status of the ansatz itself.

Problem Statement

Let $\mathcal{O}$ be a rational conformal field theory, $h$ and $v$ primary fields of $\mathcal{O}$, and IRF($\mathcal{O},h,v$) the solvable lattice model defined by Gepner's Baxterization ansatz, in which the braiding matrix exchanges the field $h$ with the field $v$ (the class is taken with $h=v$, as in the known examples): a linear combination of the projectors onto the $n$ blocks $\psi_a$ ($a=0,\dots,n-1$) of the fusion product $h\cdot h$, with coefficients built from $\sin(\zeta_j\pm u)$ of the spectral parameter $u$ and crossing parameters $\zeta_a=\frac{\pi}{2}(\Delta_{a+1}-\Delta_a)$, $\tilde{\zeta}_a=\frac{\pi}{2}(\tilde{\Delta}_{a+1}-\tilde{\Delta}_a)$ for $a=0,\dots,n-2$, where $\Delta_a$ and $\tilde{\Delta}_a$ are the conformal dimensions of the blocks of $h\cdot h$ and of $h\cdot\bar{h}$ ($\bar{h}$ the conjugate field). The problem is to place the exact solution of the thermalized (elliptic) model on rigorous footing and to complete the determination of its infrared fixed point. Its three components arise from a single computation chain — inversion relations, free energy, critical exponent, fixed point — and are jointly required.

1. Thermalization conjecture. Prove, for arbitrary models of the class, that the thermalized inversion relations hold — that the thermalized model, whose elliptic Boltzmann weights are not known in general (the thermalization being imposed at the level of the inversion relations rather than of the weights, by replacing the sine functions with $\theta_1(u,q)$), satisfies $R_i^{h,h}(u)R_i^{h,h}(-u)=\rho(u)\rho(-u)\,\mathbf{1},\qquad R_i^{h,\bar{h}}(u)R_i^{h,\bar{h}}(-u)=\tilde{\rho}(u)\tilde{\rho}(-u)\,\mathbf{1},$ with $\rho(u)=\prod_{r=0}^{n-2}\theta_1(\zeta_r-u)/\theta_1(\zeta_r)$ and $\tilde{\rho}(u)=\prod_{r=0}^{n-2}\theta_1(\tilde{\zeta}_r-u)/\theta_1(\zeta_r)$ the conjectured theta-function products ($\theta_1$ the odd Jacobi theta function, $q$ the elliptic nome measuring the distance from criticality) — or refute the conjecture by an explicit model of the class in which a thermalized inversion relation fails.

2. Analyticity and convergence. Establish, for the general class in each of the four regimes, the assumptions under which the inversion relations determine the free energy: that $\log\tilde{\kappa}(w)$ (the logarithm of the normalized largest transfer-matrix eigenvalue, as a function of $w$, with $\epsilon$ the temperature-like parameter, $w=\exp(-4\pi u/\epsilon)$ in regimes III and II and $w=\exp(2\pi u/\epsilon)$ in regimes IV and I) is analytic in the annulus containing $w=1$ and the regime's second inversion endpoint — the point $x=\exp(-4\pi\lambda/\epsilon)$ in regime III, the point $w_0$ in the other regimes (in regime II, $w_0^2=\tilde{q}^{-1}x^2$ with $x=\exp(-4\pi\lambda/\epsilon)$ and $\tilde{q}=e^{-4\pi^2/\epsilon}$; in regimes IV and I, $x=-\exp(-\pi\lambda/\epsilon)$ and $\tilde{q}=-e^{-\pi^2/\epsilon}$, with $w_0=x^{-2}$ in regime IV and $w_0=-x^{-2}\tilde{q}$ in regime I) — that the Laurent coefficients obey $d_{-m}=d_m$, and that the convergence condition $|\tilde{\zeta}_r|\leq\lambda$ holds for all $r$ (with $\lambda=\tilde{\zeta}_0=\frac{\pi}{2}\Delta_{\mathrm{adj}}$, $\Delta_{\mathrm{adj}}$ the conformal dimension of the adjoint representation of $\mathcal{O}$) — or refute them by an explicit model.

3. Identity of the coset factor. Identify the auxiliary conformal field theory $\mathcal{G}$ appearing in the fixed-point descriptions $\mathcal{C}=\mathcal{G}/\mathcal{O}$ (regimes III, IV) and $\mathcal{C}=\mathcal{O}/\mathcal{G}$ (regimes I, II) — as a construction in terms of the defining data of the model ($\mathcal{O}$, $h$, $v$ and their block structure) that reduces, in the known cases such as the ABF and affine-Lie-algebra families, to the established fixed-point identifications.

A complete answer must resolve components 1 and 2 (each either proven in the generality of the framework or refuted by a counterexample within it) and settle component 3. The coset theories $\mathcal{C}$ perturbed by the thermal operator are expected to define integrable massive quantum field theories; the systematic construction of such models is context, not a required deliverable.

References

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