Gate-controlled spin splitting in quantum dots with ferromagnetic leads in the Kondo regime

The effect of a gate voltage $({V}_{g})$ on the spin splitting of an electronic level in a quantum dot (QD) attached to ferromagnetic leads is studied in the Kondo regime using a generalized numerical renormalization group technique. We find that the ${V}_{g}$ dependence of the QD level spin splitting strongly depends on the shape of the density of states (DOS). For one class of DOS shapes there is nearly no ${V}_{g}$ dependence; for another, ${V}_{g}$ can be used to control the magnitude and sign of the spin splitting, which can be interpreted as a local exchange magnetic field. We find that the spin splitting acquires a new type of logarithmic divergence. We give an analytical explanation for our numerical results and explain how they arise due to spin-dependent charge fluctuations.