Non-equilibrium itinerant-electron magnetism: a time-dependent   mean-field theory
  We study the dynamical magnetic susceptibility of a strongly correlated electronic system in the presence of a time-dependent hopping field, deriving a generalized Bethe-Salpeter equation which is valid also out of equilibrium. Focusing on the single-orbital Hubbard model within the time-dependent Hartree-Fock approximation, we solve the equation in the non-equilibrium adiabatic regime, obtaining a closed expression for the transverse magnetic susceptibility. From this, we provide a rigorous definition of non-equilibrium (time-dependent) magnon frequencies and exchange parameters, expressed in terms of non-equilibrium single-electron Green functions and self-energies. In the particular case of equilibrium, we recover previously known results. 
Andrea Secchi, Alexander I. Lichtenstein, Mikhail I. Katsnelson
cond-mat.str-el