We develop a numerically exact scheme for resumming certain classes of Feynman diagrams in the self-consistent perturbation expansion for the electron and magnon self-energies in the nonequilibrium Green function formalism applied to a coupled electron-magnon (mbox{e-m}) system which is driven out of equilibrium by the applied finite bias voltage. Our scheme operates with the electronic and magnonic GFs and the corresponding self-energies viewed as matrices in the Keldysh space, rather than conventionally extracting their retarded and lesser components. This is employed to understand the effect of inelastic mbox{e-m} scattering on charge and spin current vs. bias voltage $V_b$ in F/I/F magnetic tunnel junctions (MTJs), which are modeled on a one-dimensional (1D) tight-binding lattice for the electronic subsystem and 1D Heisenberg model for the magnonic subsystem. For this purpose, we evaluate Fock diagram for the electronic self-energy and the electron-hole polarization bubble diagram for the magnonic self-energy. The respective electronic and magnonic GF lines within these diagrams are the fully interacting ones, thereby requiring to solve the ensuing coupled system of nonlinear integral equations self-consistently. Despite using the 1D model and treating mbox{e-m} interaction in many-body fashion only within a small active region consisting of few lattice sites around the F/I interface, our analysis captures essential features of the so-called zero-bias anomaly observed in both MgO- and AlO$_x$-based realistic 3D MTJs where the second derivative $d^2 I/dV_b^2$ (i.e., inelastic electron tunneling spectrum) of charge current exhibits sharp peaks of opposite sign on either side of the zero bias voltage.
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