Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

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.


  1. 1

    Vojta, M. Impurity quantum phase transitions. Phil. Mag. 86, 1807–1846 (2006)

    ADS  CAS  Article  Google Scholar 

  2. 2

    Bulla, R., Costi, T. A. & Pruschke, T. Numerical renormalization group method for quantum impurity systems. Rev. Mod. Phys. 80, 395–450 (2008)

    ADS  CAS  Article  Google Scholar 

  3. 3

    Hewson, A. C. The Kondo Problem to Heavy Fermions (Cambridge Univ. Press, 1997)

    Google Scholar 

  4. 4

    Cox, D. L. & Zawadowski, A. Exotic Kondo effects in metals: magnetic ions in a crystalline electric field and tunnelling centres. Adv. Phys. 47, 599–942 (1998)

    CAS  Article  Google Scholar 

  5. 5

    Dzero, M., Sun, K., Galitski, V. & Coleman, P. Topological Kondo insulators. Phys. Rev. Lett. 104, 106408 (2010)

    ADS  Article  Google Scholar 

  6. 6

    Goldhaber-Gordon, D. et al. Kondo effect in a single-electron transistor. Nature 391, 156–159 (1998)

    ADS  CAS  Article  Google Scholar 

  7. 7

    Cronenwett, S. M., Oosterkamp, T. H. & Kouwenhoven, L. P. A tunable Kondo effect in quantum dots. Science 281, 540–544 (1998)

    ADS  CAS  Article  Google Scholar 

  8. 8

    Sasaki, S., Amaha, S., Asakawa, N., Eto, M. & Tarucha, S. Enhanced Kondo effect via tuned orbital degeneracy in a spin 1/2 artificial atom. Phys. Rev. Lett. 93, 017205 (2004)

    ADS  Article  Google Scholar 

  9. 9

    Potok, R. M., Rau, I. G., Shtrikman, H., Oreg, Y. & Goldhaber-Gordon, D. Observation of the two-channel Kondo effect. Nature 446, 167–171 (2007)

    ADS  CAS  Article  Google Scholar 

  10. 10

    Kondo, J. Resistance minimum in dilute magnetic alloys. Prog. Theor. Phys. 32, 37–49 (1964)

    ADS  CAS  Article  Google Scholar 

  11. 11

    Matveev, K. A. Quantum fluctuations of the charge of a metal particle under the Coulomb blockade conditions. Sov. Phys. JETP 72, 892–899 (1991)

    Google Scholar 

  12. 12

    Yi, H. & Kane, C. L. Quantum Brownian motion in a periodic potential and the multichannel Kondo problem. Phys. Rev. B 57, R5579–R5582 (1998)

    ADS  CAS  Article  Google Scholar 

  13. 13

    Le Hur, K. Kondo resonance of a microwave photon. Phys. Rev. B 85, 140506 (2012)

    ADS  Article  Google Scholar 

  14. 14

    Goldstein, M., Devoret, M. H., Houzet, M. & Glazman, L. I. Inelastic microwave photon scattering off a quantum impurity in a Josephson-junction array. Phys. Rev. Lett. 110, 017002 (2013)

    ADS  Article  Google Scholar 

  15. 15

    Glazman, L. I. & Matveev, K. A. Lifting of the Coulomb blockade of one-electron tunneling by quantum fluctuations. Sov. Phys. JETP 71, 1031–1037 (1990)

    Google Scholar 

  16. 16

    Matveev, K. A. Coulomb blockade at almost perfect transmission. Phys. Rev. B 51, 1743–1751 (1995)

    ADS  CAS  Article  Google Scholar 

  17. 17

    Furusaki, A. & Matveev, K. A. Theory of strong inelastic cotunneling. Phys. Rev. B 52, 16676–16695 (1995)

    ADS  CAS  Article  Google Scholar 

  18. 18

    Zaránd, G., Zimányi, G. T. & Wilhelm, F. Two-channel versus infinite-channel Kondo models for the single-electron transistor. Phys. Rev. B 62, 8137–8143 (2000)

    ADS  Article  Google Scholar 

  19. 19

    Le Hur, K. & Seelig, G. Capacitance of a quantum dot from the channel-anisotropic two-channel Kondo model. Phys. Rev. B 65, 165338 (2002)

    ADS  Article  Google Scholar 

  20. 20

    Lebanon, E., Schiller, A. & Anders, F. B. Coulomb blockade in quantum boxes. Phys. Rev. B 68, 041311 (2003)

    ADS  Article  Google Scholar 

  21. 21

    Nozières, P. & Blandin, A. Kondo effect in real metals. J. Phys. 41, 193–211 (1980)

    Article  Google Scholar 

  22. 22

    Pustilnik, M., Borda, L., Glazman, L. I. & von Delft, J. Quantum phase transition in a two-channel-Kondo quantum dot device. Phys. Rev. B 69, 115316 (2004)

    ADS  Article  Google Scholar 

  23. 23

    Nygard, J., Cobden, D. H. & Lindelof, P. E. Kondo physics in carbon nanotubes. Nature 408, 342–346 (2000)

    ADS  CAS  Article  Google Scholar 

  24. 24

    Park, J. et al. Coulomb blockade and the Kondo effect in single-atom transistors. Nature 417, 722–725 (2002)

    ADS  CAS  Article  Google Scholar 

  25. 25

    Liang, W., Shores, M. P., Bockrath, M., Long, J. R. & Park, H. Kondo resonance in a single-molecule transistor. Nature 417, 725–729 (2002)

    ADS  CAS  Article  Google Scholar 

  26. 26

    Jarillo-Herrero, P. et al. Orbital Kondo effect in carbon nanotubes. Nature 434, 484–488 (2005)

    ADS  CAS  Article  Google Scholar 

  27. 27

    Ralph, D. C., Ludwig, A. W. W., von Delft, J. & Buhrman, R. A. 2-channel Kondo scaling in conductance signals from 2 level tunneling systems. Phys. Rev. Lett. 72, 1064–1067 (1994)

    ADS  CAS  Article  Google Scholar 

  28. 28

    Beenakker, C. W. J. Theory of Coulomb-blockade oscillations in the conductance of a quantum dot. Phys. Rev. B 44, 1646–1656 (1991)

    ADS  CAS  Article  Google Scholar 

  29. 29

    Joyez, P., Bouchiat, V., Esteve, D., Urbina, C. & Devoret, M. H. Strong tunneling in the single-electron transistor. Phys. Rev. Lett. 79, 1349–1352 (1997)

    ADS  CAS  Article  Google Scholar 

  30. 30

    Oreg, Y. & Goldhaber-Gordon, D. Two-channel Kondo effect in a modified single electron transistor. Phys. Rev. Lett. 90, 136602 (2003)

    ADS  Article  Google Scholar 

  31. 31

    Jezouin, S. et al. Tomonaga-Luttinger physics in electronic quantum circuits. Nature Commun. 4, 1802 (2013)

    ADS  CAS  Article  Google Scholar 

  32. 32

    Jezouin, S. et al. Quantum limit of heat flow across a single electronic channel. Science 342, 601–604 (2013)

    ADS  MathSciNet  CAS  Article  Google Scholar 

  33. 33

    Pierre, F. et al. Dephasing of electrons in mesoscopic metal wires. Phys. Rev. B 68, 085413 (2003)

    ADS  Article  Google Scholar 

  34. 34

    Wellstood, F. C., Urbina, C. & Clarke, J. Hot-electron effects in metals. Phys. Rev. B 49, 5942–5955 (1994)

    ADS  CAS  Article  Google Scholar 

  35. 35

    Parmentier, F. D. et al. Strong back-action of a linear circuit on a single electronic quantum channel. Nature Phys. 7, 935–938 (2011)

    ADS  CAS  Article  Google Scholar 

  36. 36

    Sela, E., Mitchell, A. K. & Fritz, L. Exact crossover Green function in the two-channel and two-impurity Kondo models. Phys. Rev. Lett. 106, 147202 (2011)

    ADS  Article  Google Scholar 

  37. 37

    Goldhaber-Gordon, D. et al. From the Kondo regime to the mixed-valence regime in a single-electron transistor. Phys. Rev. Lett. 81, 5225–5228 (1998)

    ADS  CAS  Article  Google Scholar 

  38. 38

    le Sueur, H. et al. Energy relaxation in the integer quantum Hall regime. Phys. Rev. Lett. 105, 056803 (2010)

    ADS  CAS  Article  Google Scholar 

  39. 39

    Fendley, P., Ludwig, A. W. W. & Saleur, H. Exact nonequilibrium transport through point contacts in quantum wires and fractional quantum Hall devices. Phys. Rev. B 52, 8934–8950 (1995)

    ADS  CAS  Article  Google Scholar 

  40. 40

    Safi, I. & Saleur, H. One-channel conductor in an ohmic environment: mapping to a Tomonaga-Luttinger liquid and full counting statistics. Phys. Rev. Lett. 93, 126602 (2004)

    ADS  CAS  Article  Google Scholar 

Download references


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




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.

Corresponding author

Correspondence to F. Pierre.

Ethics declarations

Competing interests

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.

PowerPoint slides

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation

Further reading


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing