## Abstract

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^{3,4,5} and can be explored with tunable nanostructures^{6,7,8,9}. 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^{4,11,12,13,14}. 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^{11,15,16,17,18,19,20}. 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}.

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## Acknowledgements

This work was supported by the ERC (ERC-2010-StG-20091028, no. 259033) and the French RENATECH network. We acknowledge E. Boulat, J. von Delft, S. De Franceschi, L. Glazman, D. Goldhaber-Gordon, K. Le Hur, A. Keller, K. Matveev, L. Peeters, P. Simon and G. Zaránd for critical reading of our manuscript and discussions.

## Author information

### Authors and Affiliations

### Contributions

Z.I. and F.P. performed the experiment. Z.I., A.A. and F.P. analysed the data. F.D.P. fabricated the sample. U.G. and A.C. grew the 2DEG. S.J. contributed to a preliminary experiment. F.P. led the project and wrote the manuscript with input from Z.I., A.A. and U.G.

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## Extended data figures and tables

### Extended Data Figure 1 Measurement schematic.

Schematic of the measurement setup, showing explicitly the nine different and simultaneously measured signals. *V*_{ij} (*i*, *j* ∈ {1, 2, p}) is the voltage measured with amplification chain *i* in response to the injected voltage *V*_{j}. Trenches etched in the 2DEG in the form of a Y can be seen through the metallic island.

### Extended Data Figure 2 QPC characterization.

**a**, **b**, Main panels, ‘intrinsic’ transmission probability across QPC_{1} (*τ*_{1}; **a**) and QPC_{2} (*τ*_{2}; **b**) measured at 11.5 mK (in the linear regime, without dc bias) by opening the QPC lateral characterization gate (see equivalent schematic in top left insets), and plotted versus the voltage applied to the split gate tuning the QPC (*V*_{qpc1,2}). The experimental transmission set points in the main text are indicated by symbols. Right insets, relative variation of the transmission probability with dc bias voltage, shifted vertically for clarity, for *τ*_{1,2} ≈ {0.06, 0.47, 0.93} from bottom to top, respectively. The larger noise in the inset of **a** (mostly visible for *τ*_{1} ≈ 0.06) is from the amplification chain. **c**, ‘Intrinsic’ conductance across one lateral characterization gate in units of *e*^{2}/*h* (*τ*_{lcg}, here adjacent to QPC1) plotted versus lateral gate voltage *V*_{lcg}. Increasing *V*_{lcg} results in the successive full opening of two electronic channels, as schematically illustrated. In practice, we close (open) the lateral characterization gates, corresponding to *τ*_{lcg} = 0 (*τ*_{lcg} = 2), by applying *V*_{lcg} ≈ −0.4 V (*V*_{lcg} = 0 V). Grey shaded areas correspond to the partial opening of one of the channels, a configuration not used in the experiment. **d**, Conductance of the QPCs measured at *T* = 22 mK versus dc voltage (continuous lines) with the adjacent lateral gate closed (*τ*_{lcg} = 0) and the lateral characterization gate opposite to the metallic island set to full transmission (*τ*_{lcg} = 2), which corresponds to the displayed schematic circuit. The low bias dips result from conductance suppression by the dynamical Coulomb blockade, while the high bias plateaus correspond to the ‘intrinsic’ transmission probabilities *τ*_{1,2} (horizontal dashed lines). **e**, ‘Intrinsic’ transmission probabilities *τ*_{1,2} at the experimental set points used in the main text, together with their relative increase Δ*τ*_{1,2}/*τ*_{1,2} between *V*_{dc1,2} = 0 and |*V*_{dc1,2}| = 50 µV. This increase is the main experimental factor of uncertainty in the determination of *τ*_{1,2}.

### Extended Data Figure 3 Coulomb diamonds.

The conductance *G*_{SET} (colour coded; brighter for larger *G*_{SET}) is displayed versus gate and dc voltages (δ*V*_{g} and *V*_{dc}, respectively), with both QPCs set to a low transmission probability. The Coulomb diamonds (darker) correspond to a charging energy *E*_{C} = *e*^{2}/2*C* ≈ *k*_{B} × 290 mK.

### Extended Data Figure 4 Reproducibility of conductance oscillations.

Shown is conductance across the hybrid SET (*G*_{SET}) when sweeping the gate voltage (*V*_{g}) across several periods for the symmetric configuration *τ*_{1} ≈ *τ*_{2} ≈ 0.93 and at base temperature *T* ≈ 11.5 mK (one period shown in Fig. 1c). The symbols display the measurements, the continuous line is the quantitative prediction of equation (9) in Methods.

### Extended Data Figure 5 Observation of ‘*in situ*’ conductances that overstep the standard quantum limit *e*^{2}/*h*.

The displayed Coulomb peaks were measured at *T* ≈ 14 mK for the asymmetric QPCs configuration (*τ*_{1} ≈ 0.77, *τ*_{2} ≈ 0.93). Two sweeps (*V*_{g} increasing and decreasing) are shown for each measurement. **a**–**c**, Symbols are the normalized transmitted signal *v*_{i,j}/2, with *i* ≠ *j* (equation (5) in Methods) versus gate voltage. Each panel displays the two reciprocal signals *v*_{i,j} and *v*_{j,i}. The vertical lines in **a** are visual markers used in **e**. **d**, Symbols are the normalized reflected signal at the probe QPC_{p} (2(1 − *v*_{pp}), corresponding to *G*_{p}*h*/*e*^{2} in the limit *G*_{p} ≪ *G*_{1,2}). **e**, Symbols are the *in situ* conductance ratio *G*_{1}/*G*_{2}, measured from both *v*_{1p}*/v*_{2p} and *v*_{p1}*/v*_{p2}. For each measurement, the two sweeps (*V*_{g} increasing and decreasing) are shown with different symbols. The black line is an average at a given δ*V*_{g}. The red line shows the value below which *G*_{2} > *e*^{2}/*h* near charge degeneracy (δ*V*_{g} ≈ 0). The error bars shown in **a**–**d** represent the statistical uncertainties (s.e.m.) calculated from 10 successive measurements.

### Extended Data Figure 6 Robustness to experimental conditions of ‘*in situ*’ conductances overstepping *e*^{2}/*h*.

Symbols display the ‘*in situ*’ conductance *G*_{2} measured at *T* ≈ 14 mK for the QPC setting *τ*_{1} ≈ 0.77 and *τ*_{2} ≈ 0.93, and under different experimental conditions. We repeatedly find *G*_{2} > *e*^{2}/*h*. **a**, The influence on *G*_{2} of the ac injection voltages *V*_{1,2,p}. The three lowest *V*_{ac} correspond to *V*_{1} = *V*_{2} = *V*_{p} = *V*_{ac}, whereas the fourth data point corresponds to *V*_{p} = *V*_{ac} with *V*_{1} = *V*_{2} = 1.15 µV_{rms}. **b**, Exploration of the influence on *G*_{2} of the coupling strength of QPC_{p}, characterized by *G*_{p}. The error bars represent the statistical uncertainties (s.e.m.) calculated from 20 or more different measurements.

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Iftikhar, Z., Jezouin, S., Anthore, A. *et al.* Two-channel Kondo effect and renormalization flow with macroscopic quantum charge states.
*Nature* **526**, 233–236 (2015). https://doi.org/10.1038/nature15384

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DOI: https://doi.org/10.1038/nature15384