Universal Fermi liquid crossover and quantum criticality in a mesoscopic system


Quantum critical systems derive their finite-temperature properties from the influence of a zero-temperature quantum phase transition1. The paradigm is essential for understanding unconventional high-Tc superconductors and the non-Fermi liquid properties of heavy fermion compounds. However, the microscopic origins of quantum phase transitions in complex materials are often debated. Here we demonstrate experimentally, with support from numerical renormalization group calculations, a universal crossover from quantum critical non-Fermi liquid behaviour to distinct Fermi liquid ground states in a highly controllable quantum dot device. Our device realizes the non-Fermi liquid two-channel Kondo state2,3, based on a spin-1/2 impurity exchange-coupled equally to two independent electronic reservoirs4. On detuning the exchange couplings we observe the Fermi liquid scale T*, at energies below which the spin is screened conventionally by the more strongly coupled channel. We extract a quadratic dependence of T* on gate voltage close to criticality, and validate an asymptotically exact description of the universal crossover between strongly correlated non-Fermi liquid and Fermi liquid states5,6.

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Figure 1: Device and model.
Figure 2: Quantum phase transitions.
Figure 3: Crossover from quantum criticality to a Fermi liquid.


  1. 1

    Sachdev, S. Quantum Phase Transitions 2nd edn (Cambridge Univ. Press, 2011)

    Google Scholar 

  2. 2

    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 

  3. 3

    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 

  4. 4

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

    Article  Google Scholar 

  5. 5

    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 

  6. 6

    Mitchell, A. K. & Sela, E. Universal low-temperature crossover in two-channel Kondo models. Phys. Rev. B 85, 235127 (2012)

    ADS  Article  Google Scholar 

  7. 7

    Gegenwart, P., Si, Q. & Steglich, F. Quantum criticality in heavy-fermion metals. Nature Phys. 4, 186–197 (2008)

    ADS  CAS  Article  Google Scholar 

  8. 8

    Coleman, P., Pépin, C., Si, Q. & Ramazashvili, R. How do Fermi liquids get heavy and die? J. Phys. Condens. Matter 13, R723–R738 (2001)

    ADS  CAS  Article  Google Scholar 

  9. 9

    Mebrahtu, H. T. et al. Quantum phase transition in a resonant level coupled to interacting leads. Nature 488, 61–64 (2012)

    ADS  CAS  Article  Google Scholar 

  10. 10

    Mebrahtu, H. T. et al. Observation of Majorana quantum critical behaviour in a resonant level coupled to a dissipative environment. Nature Phys. 9, 732–737 (2013)

    ADS  CAS  Article  Google Scholar 

  11. 11

    Jeong, H., Chang, A. M. & Melloch, M. R. The Kondo effect in an artificial quantum dot molecule. Science 293, 2221–2223 (2001)

    ADS  CAS  Article  Google Scholar 

  12. 12

    Bork, J. et al. A tunable two-impurity Kondo system in an atomic point contact. Nature Phys. 7, 901–906 (2011)

    ADS  CAS  Article  Google Scholar 

  13. 13

    Chorley, S. J. et al. Tunable Kondo physics in a carbon nanotube double quantum dot. Phys. Rev. Lett. 109, 156804 (2012)

    ADS  CAS  Article  Google Scholar 

  14. 14

    Emery, V. J. & Kivelson, S. Mapping of the two-channel Kondo problem to a resonant-level model. Phys. Rev. B 46, 10812–10817 (1992)

    ADS  CAS  Article  Google Scholar 

  15. 15

    Cox, D. L. Quadrupolar Kondo effect in uranium heavy-electron materials? Phys. Rev. Lett. 59, 1240–1243 (1987)

    ADS  CAS  Article  Google Scholar 

  16. 16

    Seaman, C. L. et al. Evidence for non-Fermi-liquid behavior in the Kondo alloy Y1−xUxPd3 . Phys. Rev. Lett. 67, 2882–2885 (1991)

    ADS  CAS  Article  Google Scholar 

  17. 17

    Besnus, M. J. et al. Specific heat and NMR of the Kondo system YbPd2Si2 . J. Magn. Magn. Mater. 76–77, 471–472 (1988)

    ADS  Article  Google Scholar 

  18. 18

    Ralph, D. C., Ludwig, A. W. W., von Delft, J. & Burhman, 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 

  19. 19

    Cichorek, T. et al. Two-channel Kondo effect in glasslike ThAsSe. Phys. Rev. Lett. 94, 236603 (2005)

    ADS  CAS  Article  Google Scholar 

  20. 20

    Yeh, S.-S. & Lin, J.-J. Two-channel Kondo effects in Al/AlOx/Sc planar tunnel junctions. Phys. Rev. B 79, 012411 (2009)

    ADS  Article  Google Scholar 

  21. 21

    Tóth, A. I., Borda, L., von Delft, J. & Zaránd, G. Dynamical conductance in the two-channel Kondo regime of a double dot system. Phys. Rev. B 76, 155318 (2007)

    ADS  Article  Google Scholar 

  22. 22

    Fabrizio, M., Gogolin, A. O. & Nozières Anderson-Yuval approach to the multichannel Kondo problem. Phys. Rev. B 51, 16088–16097 (1995)

    ADS  CAS  Article  Google Scholar 

  23. 23

    Anders, F. B., Lebanon, E. & Schiller, A. Coulomb blockade and non-Fermi-liquid behavior in quantum dots. Phys. Rev. B 70, 201306(R) (2004)

    ADS  Article  Google Scholar 

  24. 24

    Anders, F. B., Lebanon, E. & Schiller, A. Conductance in coupled quantum dots: indicator for a local quantum phase transition. In NIC Symposium Vol. 32, 191–199 (John von Neumann Institute for Computing, Jülich, 2006)

    Google Scholar 

  25. 25

    Lebanon, E., Schiller, A. & Anders, F. B. Enhancement of the two-channel Kondo effect in single-electron boxes. Phys. Rev. B 68, 155301 (2003)

    ADS  Article  Google Scholar 

  26. 26

    Affleck, I. & Ludwig, A. W. W. Exact conformal-field-theory results on the multichannel Kondo effect: single-fermion Green’s function, self-energy, and resistivity. Phys. Rev. B 48, 7297–7321 (1993)

    ADS  CAS  Article  Google Scholar 

  27. 27

    Borda, L., Fritz, L., Andrei, N. & Zaránd, G. Theory of inelastic scattering from quantum impurities. Phys. Rev. B 75, 235112 (2007)

    ADS  Article  Google Scholar 

  28. 28

    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 

  29. 29

    Le Hur, K., Simon, P. & Borda, L. Maximized orbital and spin Kondo effects in a single-electron transistor. Phys. Rev. B 69, 045326 (2004)

    ADS  Article  Google Scholar 

  30. 30

    Le Hur, K., Simon, P. & Loss, D. Transport through a quantum dot with SU(4) Kondo entanglement. Phys. Rev. B 75, 035332 (2007)

    ADS  Article  Google Scholar 

  31. 31

    Carmi, A., Oreg, Y., Berkooz, M. & Goldhaber-Gordon, D. Transmission phase shifts of Kondo impurities. Phys. Rev. B 86, 115129 (2012)

    ADS  Article  Google Scholar 

  32. 32

    Peeters, L., Keller, A. J., Umansky, V., Mahalu, D. & Goldhaber-Gordon, D. Repairing nanoscale devices using electron-beam-induced deposition of platinum. J. Vac. Sci. Technol. B 33, 051803 (2015)

    Article  Google Scholar 

  33. 33

    Pioro-Ladrière, M. et al. Origin of switching noise in GaAs/AlxGa1−xAs lateral gated devices. Phys. Rev. B 72, 115331 (2005)

    ADS  Article  Google Scholar 

  34. 34

    Kretinin, A. V. & Chung, Y. Wide-band current preamplifier for conductance measurements with large input capacitance. Rev. Sci. Instrum. 83, 084704 (2012)

    ADS  Article  Google Scholar 

  35. 35

    Altland, A. & Simons, B. D. Condensed Matter Field Theory 2nd edn (Cambridge Univ. Press, 2010)

    Google Scholar 

  36. 36

    Moca, C. P., Alex, A., von Delft, J. & Zaránd, G. SU(3) Anderson impurity model: a numerical renormalization group approach exploiting non-Abelian symmetries. Phys. Rev. B 86, 195128 (2012)

    ADS  Article  Google Scholar 

  37. 37

    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 

  38. 38

    Bolech, C. J. & Shah, N. Prediction of the capacitance line shape in two-channel quantum dots. Phys. Rev. Lett. 95, 036801 (2005)

    ADS  CAS  Article  Google Scholar 

  39. 39

    Mitchell, A. K., Logan, D. E. & Krishnamurthy, H. R. Two-channel Kondo physics in odd impurity chains. Phys. Rev. B 84, 035119 (2011)

    ADS  Article  Google Scholar 

  40. 40

    Wilson, K. G. The renormalization group: critical phenomena and the Kondo problem. Rev. Mod. Phys. 47, 773–840 (1975)

    ADS  MathSciNet  Article  Google Scholar 

  41. 41

    Legeza, Ö., Moca, C., Tóth, A., Weymann, I. & Zaránd, G. Manual for the Flexible DM-NRG Code. http://arXiv.org/abs/0809.3143v1 (2008)

  42. 42

    Pustilnik, M. & Glazman, L. I. Kondo effect in real quantum dots. Phys. Rev. Lett. 87, 216601 (2001)

    ADS  CAS  Article  Google Scholar 

  43. 43

    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 

  44. 44

    Weichselbaum, A. & von Delft, J. Sum-rule conserving spectral functions from the numerical renormalization group. Phys. Rev. Lett. 99, 076402 (2007)

    ADS  Article  Google Scholar 

  45. 45

    Tóth, A. I., Moca, C. P., Legeza, Ö. & Zaránd, G. Density matrix numerical renormalization group for non-Abelian symmetries. Phys. Rev. B 78, 245109 (2008)

    ADS  Article  Google Scholar 

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We are grateful to S. Amasha, Y. Oreg, A. Carmi, E. Sela, A. K. Mitchell and M. Heiblum for discussions; H. K. Choi, Y. Chung and J. MacArthur for electronics expertise; M. Heiblum for use of his laboratory during initial device characterization; H. Inoue, N. Ofek, O. Raslin and E. Weisz for fabrication guidance; F. B. Anders, E. Lebanon and the late A. Schiller for their calculations which guided prior experimental work; and M. Stopa for his SETE software for electrostatic quantum dot modelling. The device was fabricated in the Braun Submicron Center at the Weizmann Institute of Science, with final fabrication steps done at Stanford Nano Shared Facilities (SNSF) at Stanford University. This work was supported by the Gordon and Betty Moore Foundation grant no. GBMF3429, the Hungarian research grant OTKA K105149, the Polish National Science Centre project no. DEC-2013/10/E/ST3/00213, EU grant no. CIG-303 689, the National Science Foundation grant no. DMR-0906062, and the US-Israel BSF grant no. 2008149. A.J.K. and L.P. were supported by a Stanford Graduate Fellowship. SETE calculations were run on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University. NRG calculations were performed at the Poznań Supercomputing and Networking Center.

Author information




A.J.K., G.Z. and D.G.-G. designed the experiment. A.J.K. and L.P. performed the measurements. I.W., C.P.M. and G.Z. performed the NRG calculations. C.P.M. and I.W. contributed equally to the theoretical analysis. A.J.K., L.P., C.P.M., I.W., G.Z. and D.G.-G. analysed the data. A.J.K. designed and fabricated the devices, with e-beam lithography from D.M., using heterostructures grown by V.U. A.J.K. and L.P. wrote the paper with critical review provided by all other authors.

Corresponding author

Correspondence to D. Goldhaber-Gordon.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 SEM micrograph of a device nominally identical to the device studied.

Acceleration voltage in the SEM was 5 kV. The device is tilted 40° with respect to normal incidence.

Extended Data Figure 2 Sensitivity of T* and δP to fitting range.

Top panel, G(VSD,VBWT) at T = 20 mK, exactly as in Fig. 3. Middle and bottom panels, T* (middle) and δP (bottom) as functions of VBWT. Black points and red curves are exactly as in Fig. 3; the blue points correspond to the weighted mean of extracted T* and δP for an ensemble of fitting ranges, and error bars on the blue points correspond to the s.d. of the weighted mean.

Extended Data Figure 3 Measured (G(0, T) − G(VSD, T))/(kT)1/2 (symbols) and CFT fit (solid lines) of Fig. 2e broken out into separate panels for each T.

Temperature T in mK is shown centrally in each panel. The range in measured VSD is from −31.5 to 28.5 µV, resulting in a decreasing range on the (eVSD/kT)1/2 axis as T is increased. The single fit in Fig. 2e is plotted against the measured data for each T.

Extended Data Figure 4 Two-channel Kondo scaling.

Top left, measured G(VLP, VBWT) from Fig. 2c. Panels at right, measured (G(0, T) − G(VSD, T))/(kT)1/2 at six points on 2CK lines in the (VLP, VBWT) plane; points are indicated by coloured stars. Black lines are fits to thermally broadened spectral functions from ref. 26 with small phase shifts from potential scattering.

Extended Data Figure 5 Coulomb blockade thermometry.

a, Measured G(VSD, VLP) reveals two prominent linear features, the slopes of which are labelled as m and n. b, Measured G(VSD = 0, T) (crosses) and fits using equation (8) (lines). Every measurement is the average of 20 successive traces. c, Power-law fit of peak height to extracted electron temperature yields an exponent of −1.04(1). d, Residuals of the fit shown in c are all less than 0.001e2/h.

Extended Data Figure 6 Measurement of U.

Within each Coulomb diamond we label the number of electrons on the dot (N) as determined by charge sensing techniques. The intersection of the lines indicate U ≈ 2.9 meV.

Extended Data Figure 7 Measurement of EC.

a, Measurement scheme. G(VSD, VBWT) is measured using the grain’s own pair of measurement leads (red pads), which are isolated from the measurement leads of the dot by depleting gate BR. Gate BL is depleted to avoid shorting conductance through the channel just left of the grain. The current path is the red dashed line. The grey stars indicate ohmic contacts which are floated during measurement. b, G(VSD, VBWT) through the grain in the Coulomb blockade regime (T = 20 mK). The intersections of the lines indicate EC ≈ 160 µeV.

Extended Data Figure 8 Bounding Udg from measurements in the Coulomb blockade regime.

a, G(VSD, VLP) through the dot in the Coulomb blockade regime, with both the dot and grain formed. Here VBWT is such that the grain is Coulomb-blockaded. From the slopes of the dashed lines overlaid on the peaks in the data, we determine lever arms αLP = 0.081 and αSD = 0.194. b, G(VBWT, VLP) through the dot at zero VSD. Peaks in G correspond to Coulomb blockade on the dot being lifted; the splitting implies finite Udg. For fixed VBWT the difference in peak positions (blue horizontal bar) gives the dot–grain charging energy Udg = (e)(αLP)(ΔVLP) = 0.081 × 0.26 meV = 21 µeV. Dot–grain tunnelling is negligible in this limit. c, Conductance through the grain appears where expected given the interpretation of b. The conductance is measured with gates BL and BR depleted, measuring through the two point contacts formed by gate pairs LBB/BWB and LBT/BWT.

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Keller, A., Peeters, L., Moca, C. et al. Universal Fermi liquid crossover and quantum criticality in a mesoscopic system. Nature 526, 237–240 (2015). https://doi.org/10.1038/nature15261

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