We consider the Y-systems satisfied by the , , 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 ^iλ with the crossing parameter λ 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 equations, we solve the Y-systems for the finite-size corrections to the groundstate eigenvalue following the methods of Klmper and Pearce. The resulting expressions for c − 24Δ, where c is the central charge and Δ 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.

我们考虑由, ,顶点和循环模型满足的Y系统在圆柱体上具有扭曲边界条件的单位根。顶点模型分别是 6、15 和 Izergin-Korepin 19 顶点模型。相应的循环模型分别是密集、完全填充和稀释的 Temperley-Lieb 循环模型。对于所有三个模型，我们的重点是 e ^{i λ}的单位值的根，其中交叉参数λ对应于这些模型的主级数和对偶级数。将已知的函数方程转换为热力学 Bethe ansatz 方程形式的非线性积分方程，我们求解有限尺寸的Y系统 按照 Klmper 和 Pearce 的方法对基态特征值进行校正。使用各种对数恒等式简化了c - 24Δ的结果表达式，其中c是中心电荷，Δ 是与基态相关的共形权重。我们的分析结果与先前通过不同方法获得的结果一致。