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We study the dynamical magnetic susceptibility of a strongly correlatedelectronic system in the presence of a time-dependent hopping field, deriving ageneralized Bethe-Salpeter equation which is valid also out of equilibrium.Focusing on the single-orbital Hubbard model within the time-dependentHartree-Fock approximation, we solve the equation in the non-equilibriumadiabatic regime, obtaining a closed expression for the transverse magneticsusceptibility. From this, we provide a rigorous definition of non-equilibrium(time-dependent) magnon frequencies and exchange parameters, expressed in termsof non-equilibrium single-electron Green functions and self-energies. In theparticular case of equilibrium, we recover previously known results.

Non-equilibrium itinerant-electron magnetism: a time-dependent mean-field theory