We analyze the stability of excitonic ground states in the two-band Hubbard model with additional electron-phonon and Hunds rule couplings using a combination of mean-field and variational cluster approaches. We show that both the interband Coulomb interaction and the electron-phonon interaction will cooperatively stabilize a charge density wave (CDW) state which typifies an excitonic CDW if predominantly triggered by the effective interorbital electron-hole attraction or a phononic CDW if mostly caused by the coupling to the lattice degrees of freedom. By contrast, the Hunds rule coupling promotes an excitonic spin density wave. We determine the transition between excitonic charge and spin density waves and comment on a fixation of the phase of the excitonic order parameter that would prevent the formation of a superfluid condensate of excitons. The implications for exciton condensation in several material classes with strongly correlated electrons are discussed.
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