平面易辛模型的對偶性

Abstract

格拉斯曼是一個發展完善的數學結構，他可以用於計算微擾規範場論的散射幅度。在最近，該工具也被發現可以運用在平面易辛模型。本文討論了正交格拉斯曼的細胞結構與平面易辛模型之間的等價關係。我們提出了一種以微觀結構發法，來建立兩者之間的對應關係。由兩者之間的等效性，使我們可以引入兩種新遞歸方法來計算易辛網絡的關聯函數。第一種基於對偶變換，此種變換生成易辛模網絡屬於格拉斯曼中同一的細胞。我們可以用這種變換來解碎形晶格，其中遞歸公式成為有效耦合常數的精確重整化群方程。對於第二個，我們使用黏合方法，其中每次迭代將原始晶格的大小加倍。這導致更有效計算關聯函數，其中複雜度相對於辛模晶格點的數量呈對數比例增長。

Grassmannian is a well-developed mathematical structure to compute the scattering amplitudes of perturbative gauge theories. Recently, a new connection of this tool to the planar Ising model has been revealed. This thesis discusses the equivalence between planar Ising networks and cells in the positive orthogonal Grassmannian. We propose a microscopic construction based on amalgamation, which establishes the correspondence for any planar Ising network. The equivalence allows us to introduce two recursive methods for computing correlators of Ising networks. The first is based on duality moves, which generate networks belonging to the same cell in the Grassmannian. This leads to fractal lattices, where the recursion formulas become the exact RG equations of the effective couplings. For the second, we use amalgamation, where each iteration doubles the size of the seed lattice. This leads to an efficient way of computing the correlator where the complexity scales logarithmically with respect to the number of spin sites.