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Two-channel Kondo effect and renormalization flow with macroscopic quantum charge states


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 effect1,2, which entails the coupling of a quantum impurity to a continuum of states. It is central in highly correlated systems3,4,5 and can be explored with tunable nanostructures6,7,8,9. Although Kondo physics is usually associated with the hybridization of itinerant electrons with microscopic magnetic moments10, theory predicts that it can arise whenever degenerate quantum states are coupled to a continuum4,11,12,13,14. 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 island11,15,16,17,18,19,20. In contrast to other Kondo nanostructures, each conduction channel connecting the island to an electrode constitutes a distinct and fully tunable Kondo channel11, thereby providing unprecedented access to the two-channel Kondo effect and a clear path to multi-channel Kondo physics1,4,21,22. 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 point4,21. 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 criticality17.

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Figure 1: Hybrid metal–semiconductor single-electron transistor.
Figure 2: Observation of the ‘charge’ Kondo effect.
Figure 3: Interplay of two Kondo channels revealed by tuning the asymmetry.
Figure 4: Two-channel renormalization flow.


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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.

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Authors and Affiliations



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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Correspondence to F. Pierre.

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The authors declare no competing financial interests.

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. Vij (i, j {1, 2, p}) is the voltage measured with amplification chain i in response to the injected voltage Vj. 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 QPC1 (τ1; a) and QPC2 (τ2; 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 (Vqpc1,2). 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 τ1,2 ≈ {0.06, 0.47, 0.93} from bottom to top, respectively. The larger noise in the inset of a (mostly visible for τ1 ≈ 0.06) is from the amplification chain. c, ‘Intrinsic’ conductance across one lateral characterization gate in units of e2/h (τlcg, here adjacent to QPC1) plotted versus lateral gate voltage Vlcg. Increasing Vlcg results in the successive full opening of two electronic channels, as schematically illustrated. In practice, we close (open) the lateral characterization gates, corresponding to τlcg = 0 (τlcg = 2), by applying Vlcg ≈ −0.4 V (Vlcg = 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 (τlcg = 0) and the lateral characterization gate opposite to the metallic island set to full transmission (τlcg = 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 τ1,2 (horizontal dashed lines). e, ‘Intrinsic’ transmission probabilities τ1,2 at the experimental set points used in the main text, together with their relative increase Δτ1,2/τ1,2 between Vdc1,2 = 0 and |Vdc1,2| = 50 µV. This increase is the main experimental factor of uncertainty in the determination of τ1,2.

Extended Data Figure 3 Coulomb diamonds.

The conductance GSET (colour coded; brighter for larger GSET) is displayed versus gate and dc voltages (δVg and Vdc, respectively), with both QPCs set to a low transmission probability. The Coulomb diamonds (darker) correspond to a charging energy EC = e2/2C ≈ kB × 290 mK.

Extended Data Figure 4 Reproducibility of conductance oscillations.

Shown is conductance across the hybrid SET (GSET) when sweeping the gate voltage (Vg) across several periods for the symmetric configuration τ1 ≈ τ2 ≈ 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 e2/h.

The displayed Coulomb peaks were measured at T ≈ 14 mK for the asymmetric QPCs configuration (τ1 ≈ 0.77, τ2 ≈ 0.93). Two sweeps (Vg increasing and decreasing) are shown for each measurement. ac, Symbols are the normalized transmitted signal vi,j/2, with i ≠ j (equation (5) in Methods) versus gate voltage. Each panel displays the two reciprocal signals vi,j and vj,i. The vertical lines in a are visual markers used in e. d, Symbols are the normalized reflected signal at the probe QPCp (2(1 − vpp), corresponding to Gph/e2 in the limit GpG1,2). e, Symbols are the in situ conductance ratio G1/G2, measured from both v1p/v2p and vp1/vp2. For each measurement, the two sweeps (Vg increasing and decreasing) are shown with different symbols. The black line is an average at a given δVg. The red line shows the value below which G2 > e2/h near charge degeneracy (δVg ≈ 0). The error bars shown in ad 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 e2/h.

Symbols display the ‘in situ’ conductance G2 measured at T ≈ 14 mK for the QPC setting τ1 ≈ 0.77 and τ2 ≈ 0.93, and under different experimental conditions. We repeatedly find G2 > e2/h. a, The influence on G2 of the ac injection voltages V1,2,p. The three lowest Vac correspond to V1 = V2 = Vp = Vac, whereas the fourth data point corresponds to Vp = Vac with V1 = V2 = 1.15 µVrms. b, Exploration of the influence on G2 of the coupling strength of QPCp, characterized by Gp. 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).

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