Many-body correlations and macroscopic quantum behaviours are fascinating condensed matter problems. A powerful test-bed for the many-body concepts and methods is the Kondo effect(1,2), which entails the coupling of a quantum impurity to a continuum of states. It is central in highly correlated systems(3-5) and can be explored with tunable nanostructures(6-9). Although Kondo physics is usually associated with the hybridization of itinerant electrons with microscopic magnetic moments(10), theory predicts that it can arise whenever degenerate quantum states are coupled to a continuum(4,11-14). Here we demonstrate the previously elusive 'charge' Kondo effect in a hybrid metal-semiconductor implementation of a single-electron transistor, with a quantum pseudospin of 1/2 constituted by two degenerate macroscopic charge states of a metallic island(11,15-20). In contrast to other Kondo nanostructures, each conduction channel connecting the island to an electrode constitutes a distinct and fully tunable Kondo channel(11), thereby providing unprecedented access to the two-channel Kondo effect and a clear path to multi-channel Kondo physics(1,4,21,22). Using a weakly coupled probe, we find the renormalization flow, as temperature is reduced, of two Kondo channels competing to screen the charge pseudospin. This provides a direct view of how the predicted quantum phase transition develops across the symmetric quantum critical point(4,21). Detuning the pseudospin away from degeneracy, we demonstrate, on a fully characterized device, quantitative agreement with the predictions for the finite-temperature crossover from quantum criticality(17).
