We prove that for various impurity models, in both classical and quantum settings, the self-energy matrix is a sparse matrix with a sparsity pattern determined by the impurity sites. In the quantum setting, such a sparsity pattern has been known since Feynman. Indeed, it underlies several numerical methods for solving impurity problems, as well as many approaches to more general quantum many-body problems, such as the dynamical mean field theory. The sparsity pattern is easily motivated by a formal perturbative expansion using Feynman diagrams. However, to the extent of our knowledge, a rigorous proof has not appeared in the literature. In the classical setting, analogous considerations lead to a perhaps less known result, i.e., that the precision matrix of a Gibbs measure of a certain kind differs only by a sparse matrix from the precision matrix of a corresponding Gaussian measure. Our argument for this result mainly involves elementary algebraic manipulations and is in particular non-perturbative. Nonetheless, the proof can be robustly adapted to various settings of interest in physics, including quantum systems (both fermionic and bosonic) at zero- and finite-temperature, non-equilibrium systems, and superconducting systems.

中文翻译：

---

经典和量子杂质问题自能量的稀疏模式

我们证明，对于各种杂质模型，在经典和量子设置下，自能量矩阵都是一个稀疏矩阵，其稀疏模式由杂质位点决定。在量子环境中，自费曼以来就已经知道这种稀疏模式。实际上，它是解决杂质问题的几种数值方法的基础，以及解决更一般的量子多体问题的许多方法，例如动力学平均场论。稀疏模式很容易通过使用费曼图的形式扰动展开来激发。但是，就我们所知，文献中还没有严格的证据。在经典情况下，类似的考虑可能会导致鲜为人知的结果，即 某种吉布斯测度的精度矩阵与相应的高斯测度的精度矩阵只有一个稀疏矩阵不同。我们对此结果的争论主要涉及基本的代数运算，尤其是非扰动的。但是，该证明可以稳固地适应物理学中各种感兴趣的设置，包括零温度和有限温度下的量子系统（费米子和玻子），非平衡系统和超导系统。