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

For the twisted $A_1^{(1)}$, $A_2^{(1)}$ and $A_2^{(2)}$ lattice models, the 2021 analysis of Morin-Duchesne, Klümper and Pearce established the groundstate $1/N$ finite-size corrections and the associated dilogarithm identities, but only for two one-parameter subfamilies of rational crossing parameters — the principal and dual series. The unfinished problem is to extend this exact program to the remaining parameter values: (1) for generic (non-principal) rational $\lambda/\pi$, construct the analogs of the $A_1^{(1)}$ 'snake' Y-systems — closed functional-relation systems for the fused transfer matrices with the analyticity properties needed to derive and solve nonlinear integral equations (NLIEs) — for the twisted $A_2^{(1)}$ and $A_2^{(2)}$ models, and use them to obtain the groundstate finite-size corrections $c-24\Delta$ and the accompanying dilogarithm identities; (2) for $A_2^{(2)}$ in the non-compact regime $\pi/2<\lambda<2\pi/3$, extend the Y-system/NLIE machinery to resolve the degeneracy structure in which many transfer-matrix eigenvalues share the same $1/N$ correction and degeneracies are lifted only at order $1/[N(B+\log N)^2]$.

Background

A large class of two-dimensional lattice models is exactly solvable by the Yang-Baxter equation, a consistency condition guaranteeing an infinite commuting family of transfer matrices $T(u)$ indexed by a spectral parameter $u$. On a cylinder of circumference $N$ with a seam twist $\omega=e^{i\gamma}$, the largest eigenvalue admits the finite-size expansion $\log T(u) = -N f_{\mathrm{bulk}}(u) + \frac{\pi\sin\vartheta(u)}{6N}(c-24\Delta) + o(1/N)$, whose $1/N$ amplitude encodes the central charge $c$ and conformal weight $\Delta$ of the conformal field theory (CFT) describing the continuum scaling limit. The eigenvalues of the fused transfer matrices $T_{m,n}(u)$ (obtained by fusing elementary Boltzmann weights) satisfy functional relations known as the fusion hierarchy, the $T$-system and the $Y$-system; solving the associated nonlinear integral equations (NLIEs, descendants of the thermodynamic Bethe ansatz and Destri-de Vega equations) in the scaling limit yields the finite-size corrections and, as a by-product, dilogarithm identities — exact evaluations, in rational multiples of $\pi^2$, of sums of Rogers dilogarithms $L(x)$. The families considered here are $A_1^{(1)}$ (the dense loop model and six-vertex model), $A_2^{(1)}$ (the rank-2 analogue) and $A_2^{(2)}$ (the Izergin-Korepin nineteen-vertex model and its dilute loop version), each with a crossing parameter $\lambda$. At rational $\lambda/\pi$ (equivalently, $e^{i\lambda}$ a root of unity) the dense $A_1^{(1)}$ and dilute $A_2^{(2)}$ loop models realize the logarithmic minimal models $\mathcal{LM}(p,p')$ and the dilute logarithmic minimal models $\mathcal{DLM}(p,p')$ respectively — logarithmic CFTs, i.e. CFTs with logarithmic singularities in correlation functions arising from reducible-but-indecomposable representations. Morin-Duchesne, Klümper and Pearce (2021) completed the groundstate analysis for two one-parameter subfamilies of rational points: the principal series ($a=b-1$ for $A_2^{(2)}$ with $\lambda=\pi(b-a)/(2b)$; $p=p'-1$ for $A_1^{(1)}$, $A_2^{(1)}$ with $\lambda=\pi(p'-p)/p'$) and the dual series ($a=1-b$; $p=1$), reproducing known results such as $c-24\Delta = 1-3\gamma^2/(\pi(\pi-2\lambda))$ for $A_2^{(2)}$ in regime I and proving the attendant dilogarithm identities. For $A_1^{(1)}$, the non-principal rational points are known to be governed by the 'snake' Y-systems of Tateo and of Kuniba-Sakai-Suzuki (the continued-fraction TBA functional relations of the XXZ chain at roots of unity); for $A_2^{(1)}$ and $A_2^{(2)}$ no analogue is known. Separately, $A_2^{(2)}$ possesses a non-compact regime $\pi/2<\lambda<2\pi/3$ (regime III of Vernier-Jacobsen-Saleur), whose scaling limit is a non-compact CFT with continuous degrees of freedom and effective central charge approaching $c=2$; there the groundstate energy becomes infinitely degenerate in the scaling limit, finite-size amplitudes carry corrections of the form $1/[B+\log N]^2$, and many eigenvalues share the same $1/N$ correction with degeneracies lifted only at order $1/[N(B+\log N)^2]$.

Problem Statement

Extend the groundstate finite-size-correction and dilogarithm-identity program for the twisted $A_2^{(1)}$ and $A_2^{(2)}$ lattice models beyond the principal and dual series, to the remaining values of the crossing parameter $\lambda$:

(1) Non-principal rational points. For generic rational $\lambda/\pi$ outside the $A_2^{(2)}$ non-compact interval $\pi/2<\lambda<2\pi/3$ — i.e. beyond the principal series ($a=b-1$ for $A_2^{(2)}$ with $\lambda=\pi(b-a)/(2b)$; $p=p'-1$ for $A_2^{(1)}$ with $\lambda=\pi(p'-p)/p'$) and the dual series ($a=1-b$; $p=1$; the integer labels and admissibility conditions of both series are as defined in reference [1]) — construct the analogs of the $A_1^{(1)}$ 'snake' Y-systems of Tateo and Kuniba-Sakai-Suzuki: the closed systems of functional relations satisfied by the fused transfer matrices and auxiliary functions of the twisted $A_2^{(1)}$ and $A_2^{(2)}$ models, with the analyticity properties required to derive nonlinear integral equations, solve them in the scaling limit, and thereby obtain the $1/N$ groundstate corrections (the combination $c-24\Delta$) and establish the accompanying dilogarithm identities.

(2) Non-compact regime. For $A_2^{(2)}$ with $\pi/2<\lambda<2\pi/3$ (regime III), extend the same Y-system/NLIE machinery to compute the finite-size corrections, resolving the structure in which many transfer-matrix eigenvalues share the same $1/N$ correction and the degeneracy is lifted only at order $1/[N(B+\log N)^2]$ for a constant $B$.

These two components extend one and the same pipeline — fusion hierarchy, then Y-system, then NLIE, then dilogarithm identities (at rational $\lambda/\pi$), then finite-size amplitude — to the values of $\lambda$ excluded from the 2021 analysis. The problem is complete when the groundstate finite-size corrections and the accompanying dilogarithm identities of the twisted $A_2^{(1)}$ and $A_2^{(2)}$ models are established for all rational $\lambda/\pi$ outside the $A_2^{(2)}$ non-compact interval $\pi/2<\lambda<2\pi/3$, and the finite-size corrections of the $A_2^{(2)}$ model are established throughout the non-compact interval $\pi/2<\lambda<2\pi/3$, including the degeneracy structure in which the degeneracy is lifted only at order $1/[N(B+\log N)^2]$.

References

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