We present a mapping of various correlated multi-impurity Anderson models to a cluster model coupled to a number of effective conduction bands capturing its essential low-energy physics. The major ingredient is the complex single-particle self-energy matrix of the uncorrelated problem that encodes the influence to the host conduction band onto the dynamics of a set of correlated orbitals in a given geometry. While the real part of the self-energy matrix generates an effective hopping between the cluster orbitals, the imaginary part, or hybridization matrix, determines the coupling to the effective conduction electron bands in the mapped model. The rank of the hybridization matrix determines the number of independent screening channels of the problem, and allows the replacement of the phenomenological exhaustion criterion by a rigorous mathematical statement. This rank provides a distinction between multi-impurity models of the first kind and of the second kind. For the latter, there are insufficient screening channels available, so that a singlet ground state must be driven by the intercluster spin correlations. This classification provides a fundamental answer to the question of why ferromagnetic exchange interactions between local moments are irrelevant for the spin-compensated ground state in dilute multi-impurity models, whereas the formation of large spins competes with the Kondo scale in dense impurity arrays, without evoking a spin density wave. The low-temperature physics of three examples taken from the literature are deduced from the analytic structure of the mapped model, demonstrating the potential power of this approach. Numerical renormalization group calculations are presented for up to five-site clusters. We investigate the appearance of frustration-induced non-Fermi-liquid fixed points in the trimer, and demonstrate the existence of several critical points of Kosterlitz-Thouless type at which ferromagnetic correlations suppress the screening of an additional effective spin-$1/2$ degree of freedom.

中文翻译：

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高度相关的多杂质模型：从单杂质问题到晶格模型的交叉

我们提出了各种相关的多杂质安德森模型到群集模型的映射，该群集模型与捕获其基本的低能物理的许多有效导带耦合。主要成分是不相关问题的复杂单粒子自能矩阵，该矩阵将对主导带的影响编码为给定几何形状中一组相关轨道的动力学。当自能矩阵的实部在簇轨道之间产生有效跳变时，虚部或杂化矩阵确定了映射模型中与有效导带的耦合。杂交矩阵的等级决定了问题的独立筛选通道的数量，并允许通过严格的数学陈述替换现象学用尽标准。该等级提供了第一种和第二种多杂质模型之间的区别。对于后者，没有足够的筛选通道，因此必须通过簇间自旋相关性来驱动单重态基态。这种分类为以下问题提供了基本答案：在稀疏多杂质模型中，局部矩之间的铁磁交换相互作用为何与自旋补偿基态无关，而在稀有杂质阵列中，大自旋的形成与近藤尺度竞争，而没有引起自旋密度波。从映射模型的解析结构推导了从文献中选取的三个示例的低温物理学，展示了这种方法的潜在力量。给出了最多五个站点群集的数值重归一化组计算。我们研究了三聚体中由挫折感引起的非费米液体固定点的出现，并证明了Kosterlitz-Thouless型几个临界点的存在，在这些临界点处铁磁相关性抑制了对另一个有效自旋波的筛选。$\mathrm{1个}/2$ 自由度。