Skip to main content

Thank you for visiting nature.com. 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.

  • Article
  • Published:

Quantum simulation of an exotic quantum critical point in a two-site charge Kondo circuit

Nature Physics volume 19, pages 492–499 (2023)Cite this article

Subjects

Abstract

Tuning a material to the cusp between two distinct ground states can produce physical properties that are unlike those in either of the neighbouring phases. Advances in the fabrication and control of quantum systems have raised the tantalizing prospect of artificial quantum simulators that can capture such behaviour. A tunable array of coupled qubits should have an appropriately rich phase diagram, but realizing such a system with either tunnel-coupled semiconductor quantum dots or metal nanostructures has proven difficult. The challenge for scaling up to clusters or lattices is to ensure that each element behaves essentially identically and that the coupling between elements is uniform, while also maintaining tunability of the interactions. Here we study a nanoelectronic circuit comprising two coupled hybrid metal–semiconductor islands, combining the strengths of both materials to form a potentially scalable platform. The semiconductor component allows for controllable inter-site couplings at quantum point contacts, while the metal component’s effective continuum of states means that different sites can be made equivalent by tuning local potentials. The couplings afforded by this architecture can realize an unusual quantum critical point resulting from frustrated Kondo interactions. The observed critical behaviour matches theoretical predictions, verifying the success of our experimental quantum simulation.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Two-island charge Kondo device.
Fig. 2: Conductance line cuts between TPs.
Fig. 3: Universal conductance scaling near the TP.

Similar content being viewed by others

Data availability

All data used in this work are available in the Stanford Digital Repository at https://doi.org/10.25740/mx151nn9365. Source data are provided with this Paper.

References

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

  2. Paschen, S. & Si, Q. Quantum phases driven by strong correlations. Nat. Rev. Phys. 3, 9–26 (2021).

    Google Scholar 

  3. Anderson, P. W. Localized magnetic states in metals. Phys. Rev. 124, 41–53 (1961).

    ADS  MathSciNet  Google Scholar 

  4. Georges, A. & Sengupta, A. M. Solution of the two-impurity, two-channel Kondo model. Phys. Rev. Lett. 74, 2808–2811 (1995).

    ADS  Google Scholar 

  5. Affleck, I. Quantum impurity problems in condensed matter physics. Preprint at https://arxiv.org/abs/0809.3474 (2008).

  6. Gull, E. et al. Continuous-time Monte Carlo methods for quantum impurity models. Rev. Mod. Phys. 83, 349–404 (2011).

    ADS  Google Scholar 

  7. Meirav, U., Kastner, M. A. & Wind, S. J. Single-electron charging and periodic conductance resonances in GaAs nanostructures. Phys. Rev. Lett. 65, 771–774 (1990).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  10. Sasaki, S. et al. Kondo effect in an integer-spin quantum dot. Nature 405, 764–767 (2000).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  13. 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  Google Scholar 

  14. Buizert, C., Oiwa, A., Shibata, K., Hirakawa, K. & Tarucha, S. Kondo universal scaling for a quantum dot coupled to superconducting leads. Phys. Rev. Lett. 99, 136806 (2007).

    ADS  Google Scholar 

  15. Takada, S. et al. Transmission phase in the Kondo regime revealed in a two-path interferometer. Phys. Rev. Lett. 113, 126601 (2014).

    ADS  Google Scholar 

  16. Keller, A. J. et al. Emergent SU(4) Kondo physics in a spin–charge-entangled double quantum dot. Nat. Phys. 10, 145–150 (2013).

    Google Scholar 

  17. Mitchell, A. K., Liberman, A., Sela, E. & Affleck, I. SO(5) non-Fermi liquid in a coulomb box device. Phys. Rev. Lett. 126, 147702 (2021).

    ADS  Google Scholar 

  18. Keller, A. J. et al. Universal Fermi liquid crossover and quantum criticality in a mesoscopic system. Nature 526, 237–240 (2015).

    ADS  Google Scholar 

  19. Iftikhar, Z. et al. Two-channel Kondo effect and renormalization flow with macroscopic quantum charge states. Nature 526, 233–236 (2015).

    ADS  Google Scholar 

  20. Iftikhar, Z. et al. Tunable quantum criticality and super-ballistic transport in a ‘charge’ Kondo circuit. Science 360, 1315–1320 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  21. Landau, L. A., Cornfeld, E. & Sela, E. Charge fractionalization in the two-channel Kondo effect. Phys. Rev. Lett. 120, 186801 (2018).

    ADS  Google Scholar 

  22. Mitchell, A. K., Landau, L. A., Fritz, L. & Sela, E. Universality and scaling in a charge two-channel Kondo device. Phys. Rev. Lett. 116, 157202 (2016).

    ADS  Google Scholar 

  23. Zhang, G. et al. Nonequilibrium quantum critical steady state: transport through a dissipative resonant level. Phys. Rev. Res. 3, 013136 (2021).

    Google Scholar 

  24. Burdin, S., Georges, A. & Grempel, D. R. Coherence scale of the Kondo lattice. Phys. Rev. Lett. 85, 1048–1051 (2000).

    ADS  Google Scholar 

  25. Stafford, C. A. & Das Sarma, S. Collective Coulomb blockade in an array of quantum dots: a Mott-Hubbard approach. Phys. Rev. Lett. 72, 3590–3593 (1994).

    ADS  Google Scholar 

  26. Hensgens, T. et al. Quantum simulation of a Fermi–Hubbard model using a semiconductor quantum dot array. Nature 548, 70–73 (2017).

    ADS  Google Scholar 

  27. Pateras, A. et al. Electrode-induced lattice distortions in GaAs multi-quantum-dot arrays. J. Mater. Res. 34, 1291–1301 (2019).

    ADS  Google Scholar 

  28. Inoshita, T., Shimizu, A., Kuramoto, Y. & Sakaki, H. Correlated electron transport through a quantum dot: the multiple-level effect. Phys. Rev. B 48, 14725–14728 (1993).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  30. Jones, B. A., Varma, C. M. & Wilkins, J. W. Low-temperature properties of the two-impurity Kondo Hamiltonian. Phys. Rev. Lett. 61, 125–128 (1988).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  33. Zaránd, G., Chung, C.-H., Simon, P. & Vojta, M. Quantum criticality in a double-quantum-dot system. Phys. Rev. Lett. 97, 166802 (2006).

    ADS  Google Scholar 

  34. Jayatilaka, F. W., Galpin, M. R. & Logan, D. E. Two-channel Kondo physics in tunnel-coupled double quantum dots. Phys. Rev. B 84, 115111 (2011).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  36. Mitchell, A. K., Galpin, M. R., Wilson-Fletcher, S., Logan, D. E. & Bulla, R. Generalized Wilson chain for solving multichannel quantum impurity problems. Phys. Rev. B 89, 121105 (2014).

    ADS  Google Scholar 

  37. Mitchell, A. K., Pedersen, K. G., Hedegård, P. & Paaske, J. Kondo blockade due to quantum interference in single-molecule junctions. Nat. Commun. 8, 15210 (2017).

    ADS  Google Scholar 

  38. Averin, D. & Odintsov, A. Macroscopic quantum tunneling of the electric charge in small tunnel junctions. Phys. Lett. A 140, 251–257 (1989).

    ADS  Google Scholar 

  39. Averin, D. & Nazarov, Y. V. Virtual electron diffusion during quantum tunneling of the electric charge. Phys. Rev. Lett. 65, 2446–2449 (1990).

    ADS  Google Scholar 

  40. Child, T. et al. Entropy measurement of a strongly correlated quantum dot. Phys. Rev. Lett. 129, 227702 (2022)

  41. Han, C. et al. Fractional entropy of multichannel Kondo systems from conductance-charge relations. Phys. Rev. Lett. 128, 146803 (2022).

    ADS  Google Scholar 

  42. Karki, D. B., Boulat, E. & Mora, C. Double-charge quantum island in the quasiballistic regime. Phys. Rev. B 105, 245418 (2022).

    ADS  Google Scholar 

  43. 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  Google Scholar 

  44. Coleman, P., Ioffe, L. B. & Tsvelik, A. M. Simple formulation of the two-channel Kondo model. Phys. Rev. B 52, 6611–6627 (1995).

    ADS  Google Scholar 

  45. Maier, T. in Many-Body Physics: From Kondo to Hubbard Vol. 5 (eds Pavarini, E. et al.) Ch. 14 (Forschungszentrum Jülich, 2015).

  46. Sénéchal, D. in Many-Body Physics: From Kondo to Hubbard Vol. 5 (eds Pavarini, E. et al.) Ch. 13 (Forschungszentrum Jülich, 2015).

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

    ADS  MathSciNet  Google Scholar 

  48. Stadler, K. M., Mitchell, A. K., von Delft, J. & Weichselbaum, A. Interleaved numerical renormalization group as an efficient multiband impurity solver. Phys. Rev. B 93, 235101 (2016).

    ADS  Google Scholar 

  49. Galpin, M. R. et al. Conductance fingerprint of Majorana fermions in the topological Kondo effect. Phys. Rev. B 89, 045143 (2014).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  51. Minarelli, E. L., Rigo, J. B. & Mitchell, A. K. Linear response quantum transport through interacting multi-orbital nanostructures. Preprint at https://arxiv.org/abs/2209.01208 (2022).

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

    ADS  Google Scholar 

  53. Iftikhar, Z. et al. Primary thermometry triad at 6 mK in mesoscopic circuits. Nat. Commun. 7, 12908 (2016).

    ADS  MathSciNet  Google Scholar 

  54. Joyez, P. & Esteve, D. Single-electron tunneling at high temperature. Phys. Rev. B 56, 1848–1853 (1997).

    ADS  Google Scholar 

  55. Souquet, J.-R., Safi, I. & Simon, P. Dynamical Coulomb blockade in an interacting one-dimensional system coupled to an arbitrary environment. Phys. Rev. B 88, 205419 (2013).

    ADS  Google Scholar 

  56. Altimiras, C., Gennser, U., Cavanna, A., Mailly, D. & Pierre, F. Experimental test of the dynamical Coulomb blockade theory for short coherent conductors. Phys. Rev. Lett. 99, 256805 (2007).

    ADS  Google Scholar 

  57. Senkpiel, J. et al. Dynamical Coulomb blockade as a local probe for quantum transport. Phys. Rev. Lett. 124, 156803 (2020).

    ADS  Google Scholar 

  58. Jezouin, S. et al. Tomonaga–Luttinger physics in electronic quantum circuits. Nat. Commun. 4, 1802 (2013).

    ADS  Google Scholar 

  59. Flensberg, K., Girvin, S. M., Jonson, M., Penn, D. R. & Stiles, M. D. Quantum mechanics of the electromagnetic environment in the single-junction Coulomb blockade. Phys. Scripta 1992, 189 (1992).

    Google Scholar 

  60. Yeyati, A. L., Martin-Rodero, A., Esteve, D. & Urbina, C. Direct link between Coulomb blockade and shot noise in a quantum-coherent structure. Phys. Rev. Lett. 87, 046802 (2001).

    ADS  Google Scholar 

  61. Pekola, J. P., Holmqvist, T. & Meschke, M. Primary tunnel junction thermometry. Phys. Rev. Lett. 101, 206801 (2008).

    ADS  Google Scholar 

  62. Golubev, D. S. & Zaikin, A. D. Electron transport through interacting quantum dots in the metallic regime. Phys. Rev. B 69, 075318 (2004).

    ADS  Google Scholar 

  63. Devoret, M. H. et al. Effect of the electromagnetic environment on the Coulomb blockade in ultrasmall tunnel junctions. Phys. Rev. Lett. 64, 1824–1827 (1990).

    ADS  Google Scholar 

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

    Google Scholar 

  65. Slobodeniuk, A. O., Levkivskyi, I. P. & Sukhorukov, E. V. Equilibration of quantum Hall edge states by an Ohmic contact. Phys. Rev. B 88, 165307 (2013).

    ADS  Google Scholar 

Download references

Acknowledgements

We thank F. Pierre, I. Safi, G. Zarand, C. P. Moca, I. Weymann, P. Sriram, E. Sela, Y. Oreg, Q. Si, C. Varma and C. Mora for their scientific insights and suggestions. To make this project work, before coupling two islands we had to start by reproducing F. Pierre’s tour de force experiments on single islands of the same type. F. Pierre helped with comments on our fabrication process, measurement procedure and analysis. Discussions with C. Mora led to improvements in our analysis of the periodic Fermi-liquid scale. We acknowledge G. Zarand, C. P. Moca and I. Weymann for early discussions of the Hamiltonian and its implications. Measurement and analysis were supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under contract no. DE-AC02-76SF00515. Growth and characterization of heterostructures was supported by the French Renatech network. Theory and computation (A.K.M.) were supported by the Irish Research Council Laureate Awards 2017/2018 through grant no. IRCLA/2017/169. Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under award ECCS-2026822. Early research that established how to meet the demanding technical conditions for sample fabrication and basic measurements was supported by the National Science Foundation (NSF) under award no. 1608962. W.P. acknowledges support from the Fletcher Jones Fellowship. C.L.H. acknowledges support from the Gabilan Fellowship. L.P. acknowledges support of the Albion Walter Hewlett Fellowship.

Author information

Author notes

  1. These authors contributed equally: Winston Pouse, Lucas Peeters.

Authors and Affiliations

  1. Department of Applied Physics, Stanford University, Stanford, CA, USA

    Winston Pouse & Connie L. Hsueh

  2. Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA, USA

    Winston Pouse, Connie L. Hsueh, Marc A. Kastner & David Goldhaber-Gordon

  3. Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA, USA

    Winston Pouse, Lucas Peeters & Connie L. Hsueh

  4. Department of Physics, Stanford University, Stanford, CA, USA

    Lucas Peeters, Marc A. Kastner & David Goldhaber-Gordon

  5. Centre de Nanosciences et de Nanotechnologies (C2N), CNRS, Univ. Paris-Sud, Université Paris-Saclay, Palaiseau, France

    Ulf Gennser & Antonella Cavanna

  6. Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Marc A. Kastner

  7. School of Physics, University College Dublin, Belfield, Dublin, Ireland

    Andrew K. Mitchell

  8. Centre for Quantum Engineering, Science, and Technology, University College Dublin, Belfield, Dublin, Ireland

    Andrew K. Mitchell

Authors
  1. Winston Pouse
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lucas Peeters
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Connie L. Hsueh
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ulf Gennser
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Antonella Cavanna
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Marc A. Kastner
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Andrew K. Mitchell
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. David Goldhaber-Gordon
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

W.P. and L.P. performed the measurements. L.P. fabricated the device. A.K.M. developed the theory and carried out NRG calculations. W.P., L.P., C.L.H., M.A.K., A.K.M. and D.G.-G. analysed the data. A.C. and U.G. grew the heterostructure that hosts the 2DEG on which these samples are built. D.G.-G. supervised the project.

Corresponding authors

Correspondence to Andrew K. Mitchell or David Goldhaber-Gordon.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Takis Kontos and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Schematic illustrations of the models discussed.

a, The DCK model consists of six effectively independent spinless conduction electron reservoirs (blue for island, red for lead), described by fermionic operators fαγ and cα for α = L, R and γ = l, i. Tunneling occurs at each of the three QPCs controlled by JL,C,R. The island charging energy \({E}_{C}^{\alpha }\) correlates electrons fαi and cα on the same island either side of the metal component (black bar). b, At a TP, Eq. (9) describes the system at low temperatures. The three retained charge states of the two-island structure (denoted \(\left\vert A\right\rangle ,\left\vert B\right\rangle ,\left\vert C\right\rangle\)) are interconverted by QPC tunneling. The frustrated QCP arises when JL = JR = JC ≡ J. The conductive pathway \(\left\vert A\right\rangle \to \left\vert B\right\rangle \to \left\vert C\right\rangle\) corresponds to transport from left lead to right lead, and is illustrated with the blue arrows (the flow is reversed \(\left\vert C\right\rangle \to \left\vert B\right\rangle \to \left\vert A\right\rangle\) by changing the sign of the applied bias voltage).

Extended Data Fig. 2

SEM micrograph of nominally identical device. The acceleration voltage in the SEM was 5 kV.

Extended Data Fig. 3 Dynamical Coulomb blockade of QPC transmissions.

a. Measured QPC transmissions τR, τL as a function of a source-drain bias VSD for different QPC gate voltages. The measured transmission is extracted by measuring the series conductance when in series with the inter-island QPC and opposite island-lead QPC set to fully transmit a single channel (τC,L/R = 1). The measured transmissions clearly dip at zero bias, consistent with dynamical Coulomb blockade (DCB) behavior. The high bias behavior (VSD ≈ 50μV) recovers the ‘intrinsic’ transmission of the QPC, unrenormalized by DCB. b. DCB measurements comparing the right island-lead QPC to the inter-island QPC. It is clear there is a substantial difference in the DCB-renormalization at zero bias between the two, likely due to the device geometry. c. Comparison of measuring τR through both islands (blue lines, as in a, b) and through the adjacent plunger gate PR (red lines). While typically we would expect no significant bias dependence when measuring through PR, we in fact see DCB-like behavior. d. Comparing the two measurement pathways of c at fixed source-drain bias as a function of the QPC gate voltage. The ‘through the island’ (blue) measurements have been shifted by 9mV to account for the large capacitive cross-talk effect when switching between the two different measurement pathways. That the high bias traces match well is indicative that there is indeed DCB-renormalization of the transmission when measuring through PR. Empirically, using the zero bias, ‘through the plunger gate’ measurement of the transmission (solid red line), best captures the relevant transmissions in the Kondo interactions of our system.

Source data

Extended Data Fig. 4 Semi-universal τC values.

a, Measured inter-island transmission as a function of an applied gate voltage. The markers correspond to the inter-island transmissions used in Fig. 3a of the main text. b, Original line cuts in which the truncated data used in Fig. 3a are extracted from.

Source data

Supplementary information

Supplementary Information

Supplementary discussion and Figs. 1–7

Supplementary Data 1

Statistical source data.

Supplementary Data 2

Statistical source data.

Supplementary Data 3

Statistical source data.

Supplementary Data 4

Statistical source data.

Supplementary Data 5

Statistical source data.

Supplementary Data 6

Statistical source data.

Supplementary Data 7

Statistical source data.

Source data

Source Data Fig. 1

Statistical source data.

Source Data Fig. 2

Statistical source data.

Source Data Fig. 3

Statistical source data.

Source Data Extended Data Fig./Table 3

Statistical source data.

Source Data Extended Data Fig./Table 4

Statistical source data.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pouse, W., Peeters, L., Hsueh, C.L. et al. Quantum simulation of an exotic quantum critical point in a two-site charge Kondo circuit. Nat. Phys. 19, 492–499 (2023). https://doi.org/10.1038/s41567-022-01905-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-022-01905-4

This article is cited by

Search

Advanced search

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