We study the effects of electron doping in Mott insulators containing d 4 ions such as Ru 4+ , Os 4+ , Rh 5+ , and Ir 5+ with J = 0 singlet ground state. Depending on the strength of the spin-orbit coupling, the undoped systems are either nonmagnetic or host an unusual, excitonic magnetism arising from a condensation of the excited J = 1 triplet states of t 4 2g . We find that the interaction between J-excitons and doped carriers strongly supports ferromagnetism, converting both the nonmagnetic and antiferromagnetic phases of the parent insulator into a ferromagnetic metal, and further to a nonmagnetic metal. Close to the ferromagnetic phase, the low-energy spin response is dominated by intense paramagnon excitations that may act as mediators of a triplet pairing.PACS numbers: 75.10. Jm, 75.25.Dk, 75.30.Et, 74.10.+v A distinct feature of Mott insulators is the presence of low-energy magnetic degrees of freedom, and their coupling to doped charge carriers plays the central role in transition metal compounds [1]. In large spin systems like manganites, this coupling converts parent antiferromagnet (AF) into a ferromagnetic (FM) metal and gives rise to large magnetoresistivity effects. The doping of spin one-half compounds like cuprates and titanites, on the other hand, suppresses magnetic order and a paramagnetic (PM) metal emerges. In general, the fate of magnetism upon charge doping is dictated by spin-orbital structure of parent insulators.In compounds with an even number of electrons on the d shell, one may encounter a curious situation when the ionic ground state has no magnetic moment at all, yet they may order magnetically by virtue of low-lying magnetic levels with finite spin, if the exchange interactions are strong enough to overcome single-ion magnetic gap. The d 4 ions such as Ru 4+ , Os 4+ , Rh 5+ , Ir 5+ possess exactly this type level structure [2] due to spin-orbit coupling λ(S ·L): the spin S = 1 and orbital L = 1 moments form a nonmagnetic ground state with total J = 0 moment, separated from the excited level J = 1 by λ. A competition of the exchange and spin-orbit couplings results then in a quantum critical point (QCP) between nonmagnetic Mott insulator and magnetic order [3,4]. Since magnetic order is due to condensation of the virtual J = 1 levels and hence "soft", the amplitude (Higgs) mode is expected. The corollary of the "d 4 excitonic magnetism" [3] is the presence of magnetic QCP that does not require any special lattice geometry, and the energy scales involved are large. The recent neutron scattering data [5] √ 2, and T z = iT 0 , it can be written aswhere γ is determined by the bond direction. The model shows AF transition due to a condensation of T at a critical value K c = 6 11 λ of the interaction parameter K = 4t 2 0 /U . The degenerate T x,y,z levels split upon material-dependent lattice distortion, affecting the details of the model behavior [19]. We will consider the cubic symmetry case and make a few comments on the possible effects of the tetragonal splitting.The d 4 system...