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The many-body problem is usually approached from one of two perspectives: thefirst originates from an action and is based on Feynman diagrams, the second iscentered around a Hamiltonian and deals with quantum states and operators. Theconnection between results obtained in either way is made through spectral (orLehmann) representations, well known for two-point correlation functions. Here,we complete this picture by deriving generalized spectral representations formultipoint correlation functions that apply in all of the commonly usedmany-body frameworks: the imaginary-frequency Matsubara and the real-frequencyzero-temperature and Keldysh formalisms. Our approach separates spectral fromtime-ordering properties and thereby elucidates the relation between the threeformalisms. The spectral representations of multipoint correlation functionsconsist of partial spectral functions and convolution kernels. The former areformalism independent but system specific; the latter are system independentbut formalism specific. Using a numerical renormalization group (NRG) methoddescribed in the accompanying paper, we present numerical results for selectedquantum impurity models. We focus on the four-point vertex (effectiveinteraction) obtained for the single-impurity Anderson model and for thedynamical mean-field theory (DMFT) solution of the one-band Hubbard model. Inthe Matsubara formalism, we analyze the evolution of the vertex down to verylow temperatures and describe the crossover from strongly interacting particlesto weakly interacting quasiparticles. In the Keldysh formalism, we firstbenchmark our results at weak and infinitely strong interaction and then revealthe rich real-frequency structure of the DMFT vertex in the coexistence regimeof a metallic and insulating solution.

Multipoint correlation functions: spectral representation and numerical evaluation