Analytic one- and two-particle vertices for the Hubbard atom, taken from Phys. Rev. B 98, 235107 (2018) by Thunström et al.: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.98.235107.
Available are the functions bare_vertex, gf, chi, chi0, full_vertex, gamma, irreducible_vertex, channel_reducible_vertex, hedin.
To use these, you need to import SparseIR.jl to be able to create MatsubaraFreq objects.
using HubbardAtoms, SparseIR
U = 2.0
beta = 10.0
at = HubbardAtom(U, beta)
w = (FermionicFreq(11), FermionicFreq(-3), BosonicFreq(8))
full_vertex(MagneticChannel(), at, w)