Charging of a discrete electronic level as its energy goes below the Fermi sea has all the hallmarks of a quantum phase transition: degenerate ground state at criticality, diverging time-scale with a power-law parametric dependence, universality. For non-interacting electrons the critical exponents are trivial. They become fractional and interaction-dependent if the level occupation is associated with an electrostatic re-distribution of a nearby Fermi sea. We consider a minimalistic model with one/two levels coupled to one/two Fermi seas. Critical exponents are calculated analytically using a generalization of the Nozieres-Dominicis diagrammatic solution for the Fermi edge singularity problem. We demonstrate the emergence of a single low energy scale close to the quantum critical point. In the (a) strong and (b) weak interaction limits, this scale is identical to (a) the Kondo temperature of an orbital version of the Kondo effect, and (b) renormalized level width of an interacting resonant level model. Arguably exact analytical results are supported by extensive fRG and NRG numerical scaling. |