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Strongly correlated Josephson junction: Proximity effect in the single-layer Hubbard model

Don Rolih* and Rok Žitko†

  • *Contact author: don.rolih@ijs.si
  • †Contact author: rok.zitko@ijs.si
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Phys. Rev. B 114, 045124 – Published 24 July, 2026

DOI: https://doi.org/10.1103/6tps-wkk4

Abstract

We study the proximity effect in the Hubbard model coupled to BCS superconductors describing a single-layer strongly correlated electron system in a phase-biased Josephson junction. We find two distinct gapped solutions, a Mott-like insulating (M phase) and a proximitized superconducting phase (S phase), separated by a first-order transition with hysteresis. In the M phase, the large correlation-induced charge gap strongly suppresses the critical current, while the S phase behaves as a 0-junction, with a proximitized gap that closes for ϕ=π to yield a correlated metal. Phase bias and junction transparency can thus serve as tuning knobs to switch between conducting and insulating regimes. Working within the dynamical mean-field theory using the numerical renormalization group as the impurity solver, we associate M- and S-phase solutions with the doublet and singlet fixed points of the underlying superconducting Anderson impurity problem. We obtain detailed insight into the spectral structure on all energy scales. In the M phase, the self-energy has sub-gap resonances symmetrically located around the Fermi level resulting from the splitting of the “mid-gap pole” found in Mott insulators; this structure accounts for phase insensitivity.

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References (63)

  1. B. Keimer and J. E. Moore, The physics of quantum materials, Nat. Phys. 13, 1045 (2017).
  2. Y. Tokura, M. Kawasaki, and N. Nagaosa, Emergent functions of quantum materials, Nat. Phys. 13, 1056 (2017).
  3. L. Balents, C. R. Dean, D. K. Efetov, and A. F. Young, Superconductivity and strong correlations in moiré flat bands, Nat. Phys. 16, 725 (2020).
  4. H. Boschker and J. Mannhart, Quantum-matter heterostructures, Annu. Rev. Condens. Matter Phys. 8, 145 (2017).
  5. C. Wang, Z. Zhou, and L. Gao, Two-dimensional van der Waals superconductor heterostructures: Josephson junctions and beyond, Precis. Chem. 2, 273 (2024).
  6. J. Ma, Y. Yan, Z. Wan, L. Huang, K. Jin, J. Yin, B. Shen, Y. Zhang, G. Dong, and W. Han, Hybrid superconducting-magnetic van der Waals heterostructures: Progress and perspectives, Adv. Mater. (2025), doi: 10.1002/adma.202507866.
  7. A. Di Bernardo and E. Scheer, Van der Waals superconducting electronics: Materials, devices and circuit integration, arXiv:2601.07018.
  8. A. K. Geim and I. V. Grigorieva, van der Waals heterostructures, Nature (London) 499, 419 (2013).
  9. H. Wu, Y. Wang, Y. Xu, P. K. Sivakumar, C. Pasco, U. Filippozzi, S. S. Parkin, Y.-J. Zeng, T. McQueen, and M. N. Ali, The field-free Josephson diode in a van der Waals heterostructure, Nature (London) 604, 653 (2022).
  10. M. P. Dubbelman, H. Wu, J. Aretz, Y. Wang, C. M. Pasco, Y. Zhao, T. M. Kyrk, J. Yang, X. Xu, T. M. McQueen, M. Roesner, and M. N. Ali, Driving the field-free Josephson diode effect using Kagome Mott insulator barriers, arXiv:2512.17099.
  11. H. Kim, D. K. Morr, and R. Wiesendanger, Proximity-induced superconductivity in a 2D Kondo lattice of an f-electron-based surface alloy, Nano Lett. 24, 14139 (2024).
  12. A. V. Rozhkov and D. P. Arovas, Josephson coupling through a magnetic impurity, Phys. Rev. Lett. 82, 2788 (1999).
  13. J. A. van Dam, Y. V. Nazarov, E. P. A. M. Bakkers, S. De Franceschi, and L. P. Kouwenhoven, Supercurrent reversal in quantum dots, Nature (London) 442, 667 (2006).
  14. V. Meden, The Anderson–Josephson quantum dot—A theory perspective, J. Phys.: Condens. Matter 31, 163001 (2019).
  15. M. Imada, A. Fujimori, and Y. Tokura, Metal-insulator transitions, Rev. Mod. Phys. 70, 1039 (1998).
  16. A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Mod. Phys. 68, 13 (1996).
  17. M. Qin, T. Schäfer, S. Andergassen, P. Corboz, and E. Gull, The Hubbard model: A computational perspective, Annu. Rev. Condens. Matter Phys. 13, 275 (2022).
  18. A. Georges and W. Krauth, Numerical solution of the d=∞ Hubbard model: Evidence for a Mott transition, Phys. Rev. Lett. 69, 1240 (1992).
  19. M. J. Rozenberg, X. Y. Zhang, and G. Kotliar, Mott-Hubbard transition in infinite dimensions, Phys. Rev. Lett. 69, 1236 (1992).
  20. A. Georges and W. Krauth, Physical properties of the half-filled Hubbard model in infinite dimensions, Phys. Rev. B 48, 7167 (1993).
  21. X. Y. Zhang, M. J. Rozenberg, and G. Kotliar, Mott transition in the d=∞ Hubbard model at zero temperature, Phys. Rev. Lett. 70, 1666 (1993).
  22. O. Sakai and Y. Kuramoto, Application of the numerical renormalization group method to the Hubbard model in infinite dimensions, Solid State Commun. 89, 307 (1994).
  23. M. J. Rozenberg, G. Kotliar, and X. Y. Zhang, Mott-Hubbard transition in infinite dimensions. II, Phys. Rev. B 49, 10181 (1994).
  24. M. Potthoff and W. Nolting, Surface metal-insulator transition in the Hubbard model, Phys. Rev. B 59, 2549 (1999).
  25. J. K. Freericks, B. K. Nikolić, and P. Miller, Tuning a Josephson junction through a quantum critical point, Phys. Rev. B 64, 054511 (2001).
  26. J. K. Freericks, B. K. Nikolić, and P. Miller, Optimizing the speed of a Josephson junction with dynamical mean field theory, Int. J. Mod. Phys. B 16, 531 (2002).
  27. P. Miller and J. K. Freericks, Microscopic self-consistent theory of Josephson junctions including dynamical electron correlations, J. Phys.: Condens. Matter 13, 3187 (2001).
  28. J. K. Freericks, Dynamical mean-field theory for strongly correlated inhomogeneous multilayered nanostructures, Phys. Rev. B 70, 195342 (2004).
  29. S. Okamoto and A. J. Millis, Electronic reconstruction at an interface between a Mott insulator and a band insulator, Nature (London) 428, 630 (2004).
  30. S. Okamoto and A. J. Millis, Spatial inhomogeneity and strong correlation physics: A dynamical mean-field study of a model Mott-insulator–band-insulator heterostructure, Phys. Rev. B 70, 241104(R) (2004).
  31. G. Mazza, A. Amaricci, and M. Capone, Interface and bulk superconductivity in superconducting heterostructures with enhanced critical temperatures, Phys. Rev. B 103, 094514 (2021).
  32. F. Petocchi and M. Capone, Embedding dynamical mean-field theory for superconductivity in layered materials and heterostructures, Phys. Rev. B 93, 235125 (2016).
  33. R. W. Helmes, T. A. Costi, and A. Rosch, Kondo proximity effect: How does a metal penetrate into a Mott insulator? Phys. Rev. Lett. 101, 066802 (2008).
  34. H. Zenia, J. K. Freericks, H. R. Krishnamurthy, and T. Pruschke, Appearance of “fragile” Fermi liquids in finite-width Mott insulators sandwiched between metallic leads, Phys. Rev. Lett. 103, 116402 (2009).
  35. M.-S. Choi, M. Lee, K. Kang, and W. Belzig, Kondo effect and Josephson current through a quantum dot between two superconductors, Phys. Rev. B 70, 020502(R) (2004).
  36. C. Karrasch, A. Oguri, and V. Meden, Josephson current through a single Anderson impurity coupled to BCS leads, Phys. Rev. B 77, 024517 (2008).
  37. A. Bargerbos, M. Pita-Vidal, R. Žitko, J. Ávila, L. J. Splitthoff, L. Grünhaupt, J. J. Wesdorp, C. K. Andersen, Y. Liu, L. P. Kouwenhoven, R. Aguado, A. Kou, and B. van Heck, Singlet-doublet transitions of a quantum dot Josephson junction detected in a transmon circuit, PRX Quantum 3, 030311 (2022).
  38. S. Bag, L. Fratino, A. Camjayi, M. Civelli, and M. Rozenberg, Coupling strongly correlated electron systems to a tunable electronic reservoir, Phys. Rev. B 109, 195171 (2024).
  39. A. Toschi, M. Capone, and C. Castellani, Energetic balance of the superconducting transition across the BCS—Bose Einstein crossover in the attractive Hubbard model, Phys. Rev. B 72, 235118 (2005).
  40. J. Bauer, A. C. Hewson, and N. Dupuis, Dynamical mean-field theory and numerical renormalization group study of superconductivity in the attractive Hubbard model, Phys. Rev. B 79, 214518 (2009).
  41. R. Bulla, Zero temperature metal-insulator transition in the infinite-dimensional Hubbard model, Phys. Rev. Lett. 83, 136 (1999).
  42. R. Žitko, D. Hansen, E. Perepelitsky, J. Mravlje, A. Georges, and B. S. Shastry, Extremely correlated Fermi liquid theory meets dynamical mean-field theory: Analytical insights into the doping-driven Mott transition, Phys. Rev. B 88, 235132 (2013).
  43. H. Eisenlohr, S.-S. B. Lee, and M. Vojta, Mott quantum criticality in the one-band Hubbard model: Dynamical mean-field theory, power-law spectra, and scaling, Phys. Rev. B 100, 155152 (2019).
  44. K. G. Wilson, The renormalization group: Critical phenomena and the Kondo problem, Rev. Mod. Phys. 47, 773 (1975).
  45. R. Bulla, T. A. Costi, and T. Pruschke, Numerical renormalization group method for quantum impurity systems, Rev. Mod. Phys. 80, 395 (2008).
  46. F. B. Anders and A. Schiller, Real-time dynamics in quantum-impurity systems: A time-dependent numerical renormalization-group approach, Phys. Rev. Lett. 95, 196801 (2005).
  47. A. Weichselbaum and J. von Delft, Sum-rule conserving spectral functions from the numerical renormalization group, Phys. Rev. Lett. 99, 076402 (2007).
  48. R. Žitko and T. Pruschke, Energy resolution and discretization artifacts in the numerical renormalization group, Phys. Rev. B 79, 085106 (2009).
  49. R. Žitko, NRG Ljubljana (2021), https://github.com/rokzitko/nrgljubljana.
  50. J.-G. Liu, D. Wang, and Q.-H. Wang, Quantum impurities in channel mixing baths, Phys. Rev. B 93, 035102 (2016).
  51. F. B. Kugler, Improved estimator for numerical renormalization group calculations of the self-energy, Phys. Rev. B 105, 245132 (2022).
  52. Spectral functions are defined as A11(ω)=(−1/π)Im〈〈dσ;dσ†〉〉 and A12(ω)=(−1/π)Im〈〈d↓;d↑〉〉, where 〈〈A;B〉〉 denotes the retarded correlation function for operators A and B.
  53. The values of η are 1.89, 2.52, 3.25, 4.72, 6.34 corresponding to values of U shown in the inset of Fig. 2 from smallest to largest.
  54. D. E. Logan and M. R. Galpin, Mott insulators and the doping-induced Mott transition within DMFT: Exact results for the one-band Hubbard model, J. Phys.: Condens. Matter 28, 025601 (2016).
  55. S. Sen, P. J. Wong, and A. K. Mitchell, The Mott transition as a topological phase transition, Phys. Rev. B 102, 081110(R) (2020).
  56. Local charge, spin and pairing susceptibility are defined as χcharge(ω)=〈〈nd;nd〉〉, χspin(ω)=∑α〈〈Sz;Sz〉〉, χpair(ω)=〈〈d↓d↑;d↑†d↓†〉〉, where nd=nd↑+nd↓ and Sz=1/2(nd↑−nd↓).
  57. L. I. Glazman and K. A. Matveev, Resonant Josephson current through Kondo impurities in a tunnel barrier, JETP Lett. 49, 659 (1989).
  58. L. Pavešić, R. Aguado, and R. Žitko, Strong-coupling theory of quantum-dot Josephson junctions: Role of a residual quasiparticle, Phys. Rev. B 109, 125131 (2024).
  59. P. Zalom, D. Rolih, and R. Žitko, Andreev bound state spectroscopy of a quantum-dot-based Aharonov-Bohm interferometer with superconducting terminals, Phys. Rev. B 113, 075130 (2026).
  60. D. Rolih, Dataset for “Strongly correlated Josephson junction: Proximity effect in the single-layer Hubbard model” [Dataset], Zenodo, 2026, 10.5281/zenodo.18659749.
  61. J. K. Freericks, Transport in Multilayered Nanostructures: The Dynamical Mean-field Theory Approach (World Scientific, Singapore, 2006).
  62. E. N. Economou, Green's Functions in Quantum Physics (Springer, Berlin, 2006).
  63. D. Rolih, ChannelMixingDiscretization.jl (2025), https://github.com/donrolih/ChannelMixingDiscretization.jl.

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Phys. Rev. B 114, 045124– Published 24 July, 2026
  • Received 16 February 2026
  • Revised 6 May 2026
  • Accepted 17 June 2026
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