Analytic Continuation of Multipoint Correlation Functions: From Imaginary to Real Frequencies

Conceptually, the Matsubara formalism (MF), using imaginary frequencies, and the Keldysh formalism (KF), formulated in real frequencies, give equivalent results for equilibrium systems which are invariant under time translations. While the MF is more convenient due to its reduced complexity compared to the KF, the computation of physical observables requires the analytical continuation of functions from imaginary to real frequencies. The analytic continuation is well-known for two-point correlation functions, but for general multipoint correlators, a straightforward recipe for obtaining all Keldysh components from the MF correlator had not been formulated yet. Recently a representation of MF and KF correlation functions in terms of formalism-independent partial spectral functions and formalism-specific kernels was introduced [1]. We use this representation to formally elucidate the connection between both formalisms and show how arbitrary multipoint MF correlation functions can be analytically continued to yield all Keldysh components of the corresponding KF correlation function, and illustrate this procedure explicitly for the Hubbard atom.

[1] F. B. Kugler, S.-S. B. Lee and J. von Delft Phys. Rev. X 11, 041006 (2021)