论文标题
融合层次结构，$ t $ - 系统和$ y $ -Systems用于稀释的$ a_2^{（2）} $ loop型号
Fusion hierarchies, $T$-systems and $Y$-systems for the dilute $A_2^{(2)}$ loop models on a strip
论文作者
论文摘要
我们研究了宽度$ n $的几何形状上的稀释$ a_2^{（2）} $循环模型。已知边界条件的两个家族可以满足边界杨巴克斯特方程。将边界条件固定在条带的两端会导致四个型号。我们在这四个边界条件以及具有通用的交叉参数$λ$的情况下构建了模型及其$ t $和$ y $ - 系统的通勤传输矩阵的融合层次结构。对于$λ/π$合理的，因此$ q = -e^{4iλ} $是一个团结的根，我们证明了融合传输矩阵满足的线性关系，将融合层次结合到有限的系统中。融合关系使我们能够在自由能的大量扩展中计算两个主要术语，即散装和边界自由能。发现这些与针对小$ n $获得的数值数据一致。目前的工作补充了先前的研究（A. Morin-Duchesne，P.A。Pearce，J。Stat。Mech。（2019）），研究了具有定期边界条件的稀释度$ A_2^{（2）} $循环模型。
We study the dilute $A_2^{(2)}$ loop models on the geometry of a strip of width $N$. Two families of boundary conditions are known to satisfy the boundary Yang-Baxter equation. Fixing the boundary condition on the two ends of the strip leads to four models. We construct the fusion hierarchy of commuting transfer matrices for the model as well as its $T$- and $Y$-systems, for these four boundary conditions and with a generic crossing parameter $λ$. For $λ/π$ rational and thus $q=-e^{4iλ}$ a root of unity, we prove a linear relation satisfied by the fused transfer matrices that closes the fusion hierarchy into a finite system. The fusion relations allow us to compute the two leading terms in the large-$N$ expansion of the free energy, namely the bulk and boundary free energies. These are found to be in agreement with numerical data obtained for small $N$. The present work complements a previous study (A. Morin-Duchesne, P.A. Pearce, J. Stat. Mech. (2019)) that investigated the dilute $A_2^{(2)}$ loop models with periodic boundary conditions.
