Conjectural foundations of the inversion-relation solution for solvable IRF lattice models: thermalization, analyticity of the free energy, and the unidentified coset CFT $\mathcal{G}$

Gepner (Nucl. Phys. B 971 (2021) 115532) computed the exact free energy of solvable interaction-round-a-face (IRF) lattice models in all four regimes by combining a conformal-field-theory 'Baxterization' ansatz with thermodynamic inversion relations, reproducing Baxter's hard-hexagon free energy and predicting that the critical fixed point is a coset $C=\mathcal{G}/\mathcal{O}$ or $C=\mathcal{O}/\mathcal{G}$ of the defining conformal field theory $\mathcal{O}$ with an as-yet-unidentified CFT $\mathcal{G}$. The derivation rests on unproven conjectures: the thermalization of the inversion relations (replacing $\sin u$ by the Jacobi theta function $\theta_1(u,q)$) for arbitrary rational-CFT data; the analyticity of $\log\tilde\kappa(w)$ (respectively $\log[w^\mu\tilde\kappa(w)]$) in the annulus containing $w=1$ and a second, regime-dependent point, which includes the unproven bound $|\tilde\zeta_r|\le\lambda$; and the existence and identity of $\mathcal{G}$. Subsequent work extends the same conjectural framework without resolving any of these. The problem is to prove or refute each conjecture and to identify $\mathcal{G}$, thereby placing the exact free energies and fixed-point predictions for this large class of solvable models on rigorous footing.

Background

A two-dimensional lattice model of statistical mechanics is 'solvable' when its Boltzmann weights satisfy the Yang–Baxter equation, an algebraic factorization condition that guarantees an infinite family of commuting transfer matrices (the transfer matrix generates the partition function row by row; commutativity yields conserved quantities). An important class are the interaction-round-a-face (IRF) models, in which weights are attached to the faces (plaquettes) of a square lattice and depend on the four height variables at the corners, subject to admissibility rules. Baxter's solution of the hard-hexagon model (a gas of particles with exclusion on a triangular lattice, equivalent to an IRF model) combined this structure with inversion relations: identities stating that the transfer matrix at spectral parameter $u$ times the transfer matrix at $-u$ is proportional to the identity, which in the thermodynamic limit become functional equations for the partition function per site $\kappa$ (the free energy per site is $-k_BT\log\kappa$). Baxter solved these equations by expanding $\log\kappa$ as a two-sided Laurent series convergent in an annulus of the variable $w=e^{iu}$.

Gepner (arXiv:hep-th/9211100, 1992) proposed a 'Baxterization' ansatz that constructs trigonometric (critical) IRF Boltzmann weights directly from the data of a rational conformal field theory (RCFT — a two-dimensional conformal field theory with finitely many primary fields, e.g. a Wess–Zumino–Witten (WZW) model built on an affine Lie algebra): fix an RCFT $\mathcal{O}$ and primary fields $h,v$, and build the weights from the braiding matrix of $\mathcal{O}$ (the monodromy data governing how two primary fields are exchanged in their operator-product expansion). The model is denoted IRF($\mathcal{O},h,v$). The weights are parameterized by crossing parameters $\zeta_a=\frac{\pi}{2}(\Delta_{a+1}-\Delta_a)$ and $\tilde\zeta_a=\frac{\pi}{2}(\tilde\Delta_{a+1}-\tilde\Delta_a)$ built from the conformal dimensions of the primary fields in the fusion of $h$ and $v$, with $\lambda=\tilde\zeta_0=\pi\Delta_{\text{adjoint}}/2$. The same ansatz applies to solvable vertex models (weights on edges rather than faces).

Gepner (2021) extends this to the off-critical (elliptic) case by a 'thermalization' step: the trigonometric inversion relations, which contain $\sin u$, are conjectured to remain valid for the thermalized (off-critical) model upon replacing $\sin u\to\theta_1(u,q)$, where $\theta_1$ is Jacobi's odd theta function and $q$ is the elliptic nome ($q=0$ being the critical point). On this conjectural basis he solves the inversion relations for the free energy of a general IRF model in the four standard regimes (I: $-1<q<0$, $0>u>-d$; II: $0<q<1$, $0>u>-d$; III: $0<q^2<1$, $0<u<d$; IV: $-1<q^2<0$, $0<u<d$, with $d=\min_i\zeta_i,\tilde\zeta_i$), following Baxter's hard-hexagon method; for $\mathcal{O}=SU(2)_3$ the results reproduce Baxter's exact hard-hexagon free energy in all four regimes. From the singularity of the free energy near criticality, $\kappa_{\text{singular}}\propto|T-T_c|^{2-\alpha}$, he obtains the critical exponent $\alpha$ and the dimension $\Delta_p$ of the thermal (energy-density) perturbing operator at the critical fixed point, in a regime-dependent form: in regimes III and IV, $2-\alpha=1/\tilde\Delta_0$ and $\Delta_p=1-\tilde\Delta_0$; in regimes I and II, $2-\alpha=1/(1-\tilde\Delta_0)$ and $\Delta_p=\tilde\Delta_0$. Here $\tilde\Delta_0=\Delta_{\text{adjoint}}$, the conformal dimension of the adjoint representation, an identification the source itself states only conjecturally for models whose defining theory is a WZW (quantum-group) model. He then finds that the critical fixed-point CFT is described by a coset (the standard quotient construction of RCFTs, e.g. the unitary minimal models are $SU(2)_{k-1}\times SU(2)_1/SU(2)_k$): $C=\mathcal{G}/\mathcal{O}$ in regimes III and IV, and $C=\mathcal{O}/\mathcal{G}$ in regimes I and II, where $\mathcal{G}$ is some unknown CFT. This agrees with all previously identified cases: the Andrews–Baxter–Forrester (ABF) model IRF($SU(2)_k$,[1],[1]) flows to the unitary minimal models in regimes III/IV (Huse) and to the $SU(2)_k/U(1)$ parafermion theories in regimes I/II (Jimbo–Miwa–Okado; parafermion theories are two-dimensional critical theories whose currents carry fractional spin, generalizing the fermion of the Ising model), and IRF($SU(N)_k$, fundamental) flows in regime III to the $SU(N)_{k-1}\times SU(N)_1/SU(N)_k$ coset (Jimbo–Miwa–Okado).

Three load-bearing steps of this derivation are, however, explicitly conjectural or assumed, and the fixed-point statement is incomplete. First, the thermalization of the inversion relations is verified only for models whose elliptic Boltzmann weights are explicitly known. Second, the solution of the free-energy functional equations assumes that $\log\tilde\kappa(w)$ (regime III) or $\log[w^\mu\tilde\kappa(w)]$ (regimes II and IV; regime I proceeds analogously) is analytic in an annulus containing the point $w=1$ and a second, regime-dependent point ($w=x$ in regime III; the regime-specific point $w_0$ in the other regimes) — an assumption justified only by explicit examples — and this includes the bound $|\tilde\zeta_r|\le\lambda$ on the crossed-channel crossing parameters, which is 'checked in many models, but we do not have a general proof for this fact'. Third, the auxiliary CFT $\mathcal{G}$ entering the coset descriptions is unidentified, so the infrared fixed point is specified only up to an unknown factor theory. The source additionally notes that the coset theories $C$, when perturbed by their thermal operator, may define new integrable massive quantum field theories, deferring any systematic construction and classification of such models to future work.

Problem Statement

Put the inversion-relation solution of the free energy of solvable IRF lattice models on rigorous footing and complete the identification of the infrared fixed points, as posed by the open issues of Gepner (Nucl. Phys. B 971 (2021) 115532). For the general model IRF($\mathcal{O},h,v$) built from an arbitrary rational conformal field theory $\mathcal{O}$ and primary fields $h,v$:

(a) Thermalization: prove, or refute by an explicit counterexample, the conjecture that the thermalized (elliptic, off-critical) Boltzmann weights satisfy the inversion relations of the source, eqs. (2.22)–(2.25) — i.e. that the trigonometric inversion relations with $\sin u$ replaced by the Jacobi theta function $\theta_1(u,q)$ hold for arbitrary RCFT data, with $\rho(u)=\prod_{r=0}^{n-2}\theta_1(\zeta_r-u)/\theta_1(\zeta_r)$ and the analogous crossed-channel $\tilde\rho(u)$.

(b) Analyticity and convergence: prove, or refute, the analyticity assumption underlying the source's Laurent-series solution for the free energy in each of the four regimes: that in regime III $\log\tilde\kappa(w)$ is analytic in the annulus containing the points $w=1$ and $w=x$; and that in regimes II and IV $\log[w^\mu\tilde\kappa(w)]$, and in regime I $\log\tilde\kappa(w)$, is analytic in an annulus containing the point $w=1$ and the regime-specific point $w_0$ ($w_0^2=\tilde q^{-1}x^2$ in regime II; $w_0=x^{-2}$ in regime IV; $w_0=-x^{-2}\tilde q$ in regime I) — including in particular a general proof of the bound $|\tilde\zeta_r|\le\lambda$ on the crossed-channel crossing parameters, required for convergence.

(c) Fixed point: identify the CFT $\mathcal{G}$ (determining whether a single CFT can fill both coset roles across all four regimes, or whether the identification is regime-pair dependent) such that the critical fixed point of regimes III and IV is the coset $C=\mathcal{G}/\mathcal{O}$ and of regimes I and II is $C=\mathcal{O}/\mathcal{G}$ — consistently reproducing, in each regime, the thermal-operator dimensions and exponent relations derived in the source (regimes III and IV: $2-\alpha=1/\tilde\Delta_0$, $\Delta_p=1-\Delta_{\text{adjoint}}$; regimes I and II: $2-\alpha=1/(1-\tilde\Delta_0)$, $\Delta_p=\Delta_{\text{adjoint}}$; with $\tilde\Delta_0=\Delta_{\text{adjoint}}$), and the known families (ABF minimal models and parafermions, $SU(N)$ cosets) as special cases — or prove that no such $\mathcal{G}$ exists in general and delimit the class of RCFTs for which the coset description holds.

The common objective of (a)–(c) is to convert the source's conditional derivation chain (thermalized inversion relations → functional equations for $\kappa$ → annulus solution → free energy → critical exponent $\alpha$ → thermal-operator dimension → coset fixed point) into an unconditional one; a failure of any ingredient by counterexample refutes the corresponding conjecture and delimits the method. The problem is complete when each of (a), (b), (c) is settled by rigorous proof or by explicit counterexample/delimitation. The source additionally proposes, as an explicitly open-ended extension beyond this completion boundary, the systematic construction and classification of integrable massive quantum field theories obtained by perturbing the coset theories $C$ by their thermal operator; this extension is outside the scope of this task.

References

1. D. Gepner, On the free energy of solvable lattice models, Nuclear Physics B 971 (2021) 115532. DOI: 10.1016/j.nuclphysb.2021.115532, arXiv:2011.14108. https://arxiv.org/abs/2011.14108
2. V. Belavin, D. Gepner, J. Ramos Cabezas, The crossing multiplier for solvable lattice models, Journal of Physics Communications 6 (2022) 085001. DOI: 10.1088/2399-6528/ac8392, arXiv:2110.09798. https://arxiv.org/abs/2110.09798
3. J. Ramos, V. Belavin, D. Gepner, A large family of IRF solvable lattice models based on WZW models, Proceedings of Science, PoS(ICHEP2022) 432. DOI: 10.22323/1.414.0432. https://doi.org/10.22323/1.414.0432
4. V. Belavin, D. Gepner, J. Ramos Cabezas, B. Runov, On different approaches to integrable lattice models, Journal of Physics A: Mathematical and Theoretical 57 (2024) 155201. DOI: 10.1088/1751-8121/ad33dc, arXiv:2308.10329. https://arxiv.org/abs/2308.10329
5. V. Belavin, D. Gepner, J. Ramos Cabezas, B. Runov, On different approaches to IRF lattice models. Part II, Journal of High Energy Physics 02 (2025) 200. DOI: 10.1007/JHEP02(2025)200, arXiv:2409.05637. https://arxiv.org/abs/2409.05637
6. Y. Fukusumi, S. Nakashiba, Characterizing gapped phases by smeared boundary conformal field theories: Duality in unusual ordering with spontaneously broken generalized symmetries, arXiv:2605.07734 (2026). https://arxiv.org/abs/2605.07734
7. D. A. Huse, Exact exponents for infinitely many new multicritical points, Physical Review B 30 (1984) 3908. DOI: 10.1103/PhysRevB.30.3908. https://doi.org/10.1103/PhysRevB.30.3908
8. G. E. Andrews, R. J. Baxter, P. J. Forrester, Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities, Journal of Statistical Physics 35 (1984) 193. DOI: 10.1007/BF01014383. https://doi.org/10.1007/BF01014383
9. M. Jimbo, T. Miwa, M. Okado, Solvable lattice models with broken symmetry and Hecke's indefinite modular forms, Nuclear Physics B 275 (1986) 517. DOI: 10.1016/0550-3213(86)90611-5. https://doi.org/10.1016/0550-3213(86)90611-5
10. M. Jimbo, T. Miwa, M. Okado, Local state probabilities of solvable lattice models: an A_{n-1}^{(1)} family, Nuclear Physics B 300 (1988) 74. DOI: 10.1016/0550-3213(88)90587-1. https://doi.org/10.1016/0550-3213(88)90587-1
11. R. J. Baxter, Hard hexagons: exact solution, Journal of Physics A: Mathematical and General 13 (1980) L61. DOI: 10.1088/0305-4470/13/3/007. https://doi.org/10.1088/0305-4470/13/3/007
12. D. Gepner, Foundations of Rational Quantum Field Theory, I, arXiv:hep-th/9211100 (1992). https://arxiv.org/abs/hep-th/9211100
13. D. Gepner, B spin vertex models and quantum algebras, Nuclear Physics B 958 (2020) 115116. DOI: 10.1016/j.nuclphysb.2020.115116. https://doi.org/10.1016/j.nuclphysb.2020.115116