标题： 非阿贝尔自旋模型的对偶公式：局域表示和低温渐近性 显示英文标题

标题： Dual formulations of non-abelian spin models: local representation and low-temperature asymptotics

摘要： 具有对称群 SU(N) 或 U(N) 的非阿贝尔晶格自旋模型可以用满足比安基约束的链路变量来表述。利用这种表示法，我们推导出了这些模型在任意维度 D 的立方晶格上的配分函数的精确且局域的对偶形式。局域性意味着对偶作用由对偶晶格的某些超立方体子集上的求和给出，并且对偶变量之间的相互作用限制在一个给定的超立方体内。 对偶变量通常是离散值，并且生活在对偶晶格的 (D-2)-胞上，与 XY 模型非常相似。我们利用这一构建方法详细研究了二维 SU(2) 主要色荷模型的对偶形式。我们也给出了任意表示下的两点关联函数以及缺陷自由能的对偶表达式。 对偶玻尔兹曼因子的渐近展开的主导项被计算出来，并证明在低温下它会均匀地收敛到某种高斯分布的所有对偶变量波动中。这一结果使我们能够定义对偶形式的半经典极限，并确定配分函数的涡旋-旋波表示的类似物。 这种表示用于提取关联函数的主要微扰贡献，该函数在弱耦合下表现出幂律衰减。我们还提出了一些分析证据，表明对偶形式的低温极限完全由 SU(2) 矩阵元的 ISO(2)-类似逼近描述。 显示英文摘要

摘要： Non-abelian lattice spin models with symmetry group SU(N) or U(N) can be formulated in terms of link variables which are subject to the Bianchi constraints. Using this representation we derive exact and local dual formulation for the partition function of such models on a cubic lattice in arbitrary dimension D. Locality means that the dual action is given by a sum over some subset of hypercubes of the dual lattice and the interaction between dual variables ranges over one given hypercube. Dual variables are in general discrete-valued and live on (D-2)-cell of the dual lattice, in close analogy with the XY model. We use our construction to study in details the dual of SU(2) principal chiral model in two dimensions. We give dual expressions also for two-point correlation function in arbitrary representation and for the free energy of defects. Leading terms of the asymptotic expansion of the dual Boltzmann factor are computed and it is proven that at low temperatures it converges to a certain Gaussian distribution uniformly in all fluctuations of dual variables. This result enables us to define the semiclassical limit of the dual formulation and to determine an analog of the vortex--spin-wave representation for the partition function. Such representation is used to extract leading perturbative contribution to the correlation function which shows power-like decay at weak coupling. We also present some analytical evidences that the low-temperature limit of the dual formulation is completely described by ISO(2)-like approximation of SU(2) matrix elements.