Tensor network methods applied to the computation kernel approach for diagrammatic expansions

By merging algorithmic Matsubara integration with discrete pole representations a procedure exists to generate fully analytic closed form results for impurity problems at fixed perturbation order. By projecting the single-particle Green's function to an auxiliary space, we show how one can convert an arbitrary Feynman graph to a universal kernel representation. Once constructed, the computation kernel contains no problem-specific information yet contains all explicit temperature and frequency dependence of the diagram. We contrast the scaling properties of naive diagram evaluation for Green's functions with off-diagonal elements and demonstrate that the computation kernel approach leads to a new class of Feynman diagrams that are equivalent to tensor networks. Leveraging tensor network methodologies we discuss the improved scaling properties and give benchmarks where available.