带上稀疏$A_2^{(2)}$环模型的融合层次、$T$ -系和PDMATHAAB-系
Fusion hierarchies, $T$-systems and $Y$-systems for the dilute $A_2^{(2)}$ loop models on a strip
摘要：
研究了宽度为$N$的带状几何上的稀$A_2^{(2)}$圈模型.已知满足边界Yang-Baxter方程的两类边界条件。将边界条件固定在带的两端，得到四种模型。我们构造的融合层次的交换传递矩阵的模型，以及它的$T$ -和$Y$ -系统，这四个边界条件和通用的交叉参数$λ$。对于$λ/π$合理的，因此$q=-e^{4iλ}$的一个根的单位，我们证明了一个线性关系所满足的融合转移矩阵，关闭融合层次到一个有限的系统。融合关系使我们能够计算两个领先的条款在大$N$扩展的自由能，即体积和边界自由能。这些被发现是在协议与数值数据获得的小$N$。本文的工作是对前人研究（A. Morin-Duchesne，P.A. Pearce，J. Stat. Mech.（2019）），研究了具有周期性边界条件的稀释$A_2^{(2)}$回路模型。
英文摘要：
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.
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