  We review here some universal aspects of the physics of two-electron
molecular transistors in the absence of strong spin-orbit effects. Several
recent quantum dots experiments have shown that an electrostatic backgate could
be used to control the energy dispersion of magnetic levels. We discuss how the
generically asymmetric coupling of the metallic contacts to two different
molecular orbitals can indeed lead to a gate-tunable Hund's rule in the
presence of singlet and triplet states in the quantum dot. For gate voltages
such that the singlet constitutes the (non-magnetic) ground state, one
generally observes a suppression of low voltage transport, which can yet be
restored in the form of enhanced cotunneling features at finite bias. More
interestingly, when the gate voltage is controlled to obtain the triplet
configuration, spin S=1 Kondo anomalies appear at zero-bias, with non-Fermi
liquid features related to the underscreening of a spin larger than 1/2.
Finally, the small bare singlet-triplet splitting in our device allows to
fine-tune with the gate between these two magnetic configurations, leading to
an unscreening quantum phase transition. This transition occurs between the
non-magnetic singlet phase, where a two-stage Kondo effect occurs, and the
triplet phase, where the partially compensated (underscreened) moment is akin
to a magnetically "ordered" state. These observations are put theoretically
into a consistent global picture by using new Numerical Renormalization Group
simulations, taylored to capture sharp finie-voltage cotunneling features
within the Coulomb diamonds, together with complementary out-of-equilibrium
diagrammatic calculations on the two-orbital Anderson model. This work should
shed further light on the complicated puzzle still raised by multi-orbital
extensions of the classic Kondo problem.
