Abstract
The many-body problem is usually approached from one of two perspectives: the first originates from an action and is based on Feynman diagrams, the second is centered around a Hamiltonian and deals with quantum states and operators. The connection between results obtained in either way is made through spectral (or Lehmann) representations, well known for two-point correlation functions. Here, we complete this picture by deriving generalized spectral representations for multipoint correlation functions that apply in all of the commonly used many-body frameworks: the imaginary-frequency Matsubara and the real-frequency zero-temperature and Keldysh formalisms. Our approach separates spectral from time-ordering properties and thereby elucidates the relation between the three formalisms. The spectral representations of multipoint correlation functions consist of partial spectral functions and convolution kernels. The former are formalism independent but system specific; the latter are system independent but formalism specific. Using a numerical renormalization group (NRG) method described in the accompanying paper, we present numerical results for selected quantum impurity models. We focus on the four-point vertex (effective interaction) obtained for the single-impurity Anderson model and for the dynamical mean-field theory (DMFT) solution of the one-band Hubbard model. In the Matsubara formalism, we analyze the evolution of the vertex down to very low temperatures and describe the crossover from strongly interacting particles to weakly interacting quasiparticles. In the Keldysh formalism, we first benchmark our results at weak and infinitely strong interaction and then reveal the rich real-frequency structure of the DMFT vertex in the coexistence regime of a metallic and insulating solution.
Highlights
IV–V of that paper for how partial spectral functions (PSFs) are computed with numerical renormalization group (NRG) as sums of discrete δ peaks, and to
We show the final results of the 4p vertex, for the Mahan impurity model (MIM) in the zero-temperature formalism (ZF) and for the Anderson impurity model (AIM), with both a boxed-shaped and a dynamical mean-field theory (DMFT) self-consistent hybridization, in the Matsubara formalism (MF) and Keldysh formalism (KF)
Most numerical work involving nonperturbative 4p functions is obtained in the MF, where, thanks to the steady progress in quantum Monte Carlo (QMC) techniques, local 4p functions can nowadays be computed with high precision
Summary
A major element in the ongoing challenge of the quantum many-body problem is to extend our understanding, our analytical and numerical control, from the single- to the many-particle level. Almost since the beginning of interest in the many-body problem, two-particle correlation functions have played an important role They describe the effective interaction between two particles in the many-body environment, response functions to optical or magnetic probes, collective modes, bound states, and. The numerical analytic continuation from imaginary to real frequencies is an illconditioned problem [15]; though it may work fairly well for docile 2p functions, it is unfeasible for higher-point objects. It is of great interest to extend the understanding of lp functions that has been reached for imaginary frequencies to the real(-frequency) world
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