Groundstate finite-size corrections and dilogarithm identities for the twisted $A_1^{(1)}$, $A_2^{(1)}$ and $A_2^{(2)}$ models

TL;DR

This paper analyzes finite-size corrections and dilogarithm identities for specific twisted $A_1^{(1)}$, $A_2^{(1)}$, and $A_2^{(2)}$ models at roots of unity, providing new analytic results consistent with prior findings.

Contribution

It introduces a method to compute finite-size groundstate eigenvalue corrections for these models using TBA equations and dilogarithm identities, including new results for the dual $A_2^{(1)}$ series.

Findings

Derived explicit formulas for $c-24\,\Delta$ using dilogarithm identities.

Confirmed consistency with previous results for known models.

Provided new analytic results for the dual $A_2^{(1)}$ model.

Abstract

We consider the $Y$-systems satisfied by the $A_1^{(1)}$, $A_2^{(1)}$, $A_2^{(2)}$ vertex and loop models at roots of unity with twisted boundary conditions on the cylinder. The vertex models are the 6-, 15- and Izergin-Korepin 19-vertex models respectively. The corresponding loop models are the dense, fully packed and dilute Temperley-Lieb loop models respectively. For all three models, our focus is on roots of unity values of $e^{i\lambda}$ with the crossing parameter $\lambda$ corresponding to the principal and dual series of these models. Converting the known functional equations to nonlinear integral equations in the form of Thermodynamic Bethe Ansatz (TBA) equations, we solve the $Y$-systems for the finite-size $\frac 1N$ corrections to the groundstate eigenvalue following the methods of Kl\"umper and Pearce. The resulting expressions for $c-24\Delta$, where $c$ is the central charge and $\Delta$ is the conformal weight associated with the groundstate, are simplified using various dilogarithm identities. Our analytic results are in agreement with previous results obtained by different methods and are new for the dual series of the $A_2^{(1)}$ model.