Non-principal ('snake') $Y$-systems and exact finite-size corrections for the twisted $A_2^{(1)}$ and $A_2^{(2)}$ lattice models

The twisted $A_2^{(1)}$ (15-vertex and fully-packed loop) and $A_2^{(2)}$ (Izergin-Korepin 19-vertex and dilute loop) lattice models are Yang-Baxter integrable, and the $\frac1N$ finite-size correction of their transfer-matrix eigenvalues on a cylinder yields the conformal data combination $c-24\Delta$. Morin-Duchesne, Klümper and Pearce (2021) computed these corrections by converting the models' roots-of-unity $Y$-systems into thermodynamic Bethe ansatz type nonlinear integral equations and evaluating them with Rogers dilogarithm identities — but only on the principal and dual series of rational crossing parameters $\lambda/\pi$. Extending to non-principal roots of unity is the open core of their program: for $A_1^{(1)}$ this extension is governed by the rank-one 'snake' $Y$-systems of Tateo and of Kuniba-Sakai-Suzuki, whereas for $A_2^{(1)}$ and $A_2^{(2)}$ the analogous functional systems are still not known, and the roots-of-unity $Y$-systems previously derived for these families lack the analyticity properties required to compute conformal eigenenergies. The problem is to construct these non-principal $Y$-systems together with their analyticity and periodicity structure, solve them by the nonlinear-integral-equation/thermodynamic-Bethe-ansatz method, prove the associated dilogarithm identities, and extract the exact groundstate conformal data as a function of the twist, at least for prototypical non-principal values. Subsequent work (the 2023 site-percolation analysis via $T$-$Q$ relations, and the 2025 conjectural modular-covariant partition functions) bypassed rather than built the required $Y$-systems, leaving the problem open.

Background

Two-dimensional lattice models are exactly solvable when their Boltzmann weights satisfy the Yang-Baxter equation; the resulting commuting transfer matrices $T(u)$ depend on a spectral parameter $u$ and a crossing parameter $\lambda$. The three simplest families are classified by the affine Lie algebras $A_1^{(1)}$, $A_2^{(1)}$ and $A_2^{(2)}$: as vertex models they are the 6-vertex, 15-vertex and Izergin-Korepin 19-vertex models, and as loop models they are the dense, fully packed and dilute Temperley-Lieb loop models, whose non-local loop segments carry a fugacity determined by $\lambda$. On a cylinder of circumference $N$, the transfer matrices act on standard modules $\mathcal{W}_{N,d}$ labelled by the number $d$ of defects (through-lines), with a twist $\omega=e^{i\gamma}$ that tracks the winding of the defects around the cylinder.

In the continuum scaling limit the eigenvalues of $T(u)$ encode a two-dimensional conformal field theory (CFT): the free energy of an eigenvalue has an expansion whose $\frac1N$ term is proportional to $c-24\Delta$, where $c$ is the central charge and $\Delta$ the conformal weight of the associated scaling state. This finite-size-correction route to conformal data was made analytic by Klümper and Pearce: fused transfer matrices $T^{a}(u)$, obtained by iterating the fusion procedure, satisfy bilinear functional relations called $T$-systems, and suitable ratios $Y^{a}$ of fused eigenvalues satisfy multiplicative $Y$-systems of the form $Y^{a}(u+\lambda)Y^{a}(u-\lambda)=\prod_{b}(1+Y^{b}(u))^{C_{ab}}$. Given the analyticity properties of the eigenvalues in strips of the spectral parameter, these functional relations can be converted into nonlinear integral equations (NLIEs) in thermodynamic Bethe ansatz (TBA) form, whose scaling-limit solution is a sum of Rogers dilogarithm $L(x)$ integrals; exact dilogarithm identities then collapse these sums to rational multiples of $\pi^{2}$, giving $c-24\Delta$ in closed form.

At roots of unity $\lambda/\pi\in\mathbb{Q}$ the crossing parameter is parametrized as $\lambda=\frac{\pi(p'-p)}{p'}$ with $p,p'$ coprime for $A_1^{(1)}$ and $A_2^{(1)}$, and $\lambda=\frac{\pi(b-a)}{2b}$ with $\gcd(a,b)=1$ for $A_2^{(2)}$. Two one-parameter subfamilies, the principal series ($p=p'-1$; $a=b-1$) and the dual series ($p=1$; $a=1-b$), have simple truncation properties and were solved in the source paper: Morin-Duchesne, Klümper and Pearce proved the corresponding dilogarithm identities and computed $c-24\Delta$ for the twisted $A_1^{(1)}$, $A_2^{(1)}$ and $A_2^{(2)}$ groundstates, in agreement with previously known results. All remaining rational values — the non-principal series — are unsolved for the two rank-two families. In rank one the non-principal analysis exists: Tateo derived sine-Gordon $Y$-systems with new functional dilogarithm identities, and Kuniba-Sakai-Suzuki developed the continued-fraction TBA for the XXZ/six-vertex model at generic roots of unity; the source paper refers to these functional systems as 'snake' $Y$-systems. The loop models at roots of unity realize logarithmic (non-unitary) CFTs, with Coulomb-gas central charges $c=1-\frac{6(p'-p)^{2}}{pp'}$ for the dense $A_1^{(1)}$ and dilute-branch $A_2^{(2)}$ families (for the latter the $(p,p')$ labels are related to the $(a,b)$ parameters by $(p,p')=(2a,b)$ for $b$ odd and $(p,p')=(a,b/2)$ for $b$ even) and $c=2-\frac{6(p'-p)^{2}}{pp'}$ for the $A_2^{(1)}$ family, together with Kac-table conformal weights; these provide firm targets against which any new derivation is checked.

Later developments shape the current status. The application announced in the source conclusion — critical site percolation on the triangular lattice, the $\lambda\to\pi/3$ limit of the dilute $A_2^{(2)}$ loop model — was carried out by the same authors in 2023 by an alternative route: cubic functional relations and Baxter $T$-$Q$ equations were converted into NLIEs, giving the groundstate conformal weights analytically, the excitation spectrum numerically with conjectural trace scaling limits, and the cylinder and torus conformal partition functions, which coincide with those of critical bond percolation. That work states explicitly that the generalized 'snake' $Y$-systems are still not known for the $A_2^{(2)}$ model, and that the $Y$-systems previously derived at roots of unity involve auxiliary functions without the analyticity properties needed to compute conformal eigenenergies — the precise obstruction that any solution of the present problem must overcome (at the percolation point itself, the auxiliary functions realize an algebraic $Y$-system of Gliozzi and Tateo with no spectral-parameter shift). In 2025 the same authors conjectured the scaling limits of the transfer-matrix traces and the modular-covariant torus partition functions of the dense $A_1^{(1)}$ and dilute $A_2^{(2)}$ loop models at all roots of unity, supported by numerics and Coulomb-gas identities — but as conjectures, not $Y$-system derivations, and the $A_2^{(1)}$ family is not covered even conjecturally. A further regime of the dilute $A_2^{(2)}$ model, the non-compact regime $\frac{\pi}{2}<\lambda<\frac{2\pi}{3}$ where the groundstate becomes infinitely degenerate in the scaling limit, was studied by Vernier-Jacobsen-Saleur and remains a separate open question.

Problem Statement

Construct the non-principal ('snake') $Y$-systems of the twisted $A_2^{(1)}$ and $A_2^{(2)}$ vertex and loop models, and solve them to obtain the exact groundstate finite-size corrections at non-principal roots of unity. Concretely, for the $A_2^{(1)}$ models with $\lambda=\frac{\pi(p'-p)}{p'}$ and the $A_2^{(2)}$ models with $\lambda=\frac{\pi(b-a)}{2b}$, outside the principal series ($p=p'-1$; $a=b-1$), the dual series ($p=1$; $a=1-b$) and the percolation point of the dilute $A_2^{(2)}$ family ($\lambda=\frac{\pi}{3}$, i.e. $b=3a$):

(i) derive the closed system of functional equations satisfied by the fused transfer-matrix eigenvalues, equivalently by suitable $Y$-functions, at these generic rational values, together with the analyticity (zero- and pole-free strip) and periodicity properties required to convert them into nonlinear integral equations — a property the known roots-of-unity $Y$-systems of these families demonstrably lack;

(ii) carry out the TBA/NLIE analysis and evaluate the $\frac1N$ finite-size correction of the groundstate in the standard modules $\mathcal{W}_{N,d}$ as a function of the twist $\gamma$ and defect number $d$, obtaining the combination $c-24\Delta$;

(iii) state and prove the Rogers-dilogarithm identities to which the solution reduces;

(iv) verify the resulting conformal data against the known Coulomb-gas values (e.g. $c=2-\frac{6(p'-p)^{2}}{pp'}$ for $A_2^{(1)}$, $c=1-\frac{6(p'-p)^{2}}{pp'}$ on the dilute-branch $A_2^{(2)}$ family, with the two parametrizations related by $(p,p')=(2a,b)$ for $b$ odd and $(p,p')=(a,b/2)$ for $b$ even, and Kac-table weights) and against finite-$N$ numerics.

Completion requires the derivation of the functional systems for both families and their solution demonstrated at least for prototypical non-principal values of each — the level at which the source paper solved the principal and dual series. At this level items (i)–(iii) are mandatory — the functional system with its analyticity and periodicity structure, the TBA/NLIE evaluation of $c-24\Delta$ as a function of the twist $\gamma$ and defect number $d$, and the dilogarithm identities used — whereas (iv) is a consistency check that may be numerical. Since the vertex and loop realizations of a family share one $Y$-system, it suffices to derive and solve it at one non-principal value of each family, for either realization; any value in the non-principal range above qualifies. A formulation valid for all non-principal rational $\lambda/\pi$, and extension to finite excitations and conformal partition functions, are the natural continuation but not required for the core. The non-compact regime $\frac{\pi}{2}<\lambda<\frac{2\pi}{3}$ of $A_2^{(2)}$ (Regime III), where the groundstate is infinitely degenerate with splitting only at order $\frac{1}{N(B+\log N)^{2}}$, is a related but separate open question and is not part of this problem.

References

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