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arXiv:2602.14796 · Physical review. B./Physical review. B

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

Don Rolih, Rok Žitko

Published Jun 17, 2026• 1 citation
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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.

DOI
10.1103/6tps-wkk4

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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.
In this letter, we have studied the effects of proximity of a superconductor to a normal metal. The system, represented by a bilayer attractive Hubbard model, is investigated using layer-dynamical mean field theory and iterated perturbation theory for superconductivity as an impurity solver. The bilayer system comprises a superconducting and a normal metallic layer, connected by an inter-planar hopping (t⊥t_\perpt⊥​). It is found that superconductivity is induced in the normal layer for small t⊥t_\perpt⊥​. With increasing inter-layer hopping, the bilayer system undergoes two transitions: a first order transition to a normal metallic phase, and subsequently a continuous crossover to a band insulator.
Near a Mott transition, strong electron correlations may enhance Cooper pairing. This is demonstrated in the dynamical mean field theory solution of a twofold-orbital degenerate Hubbard model with an inverted on-site Hund rule exchange, favoring local spin-singlet configurations. Close to the Mott insulator (which here is a local version of a valence bond insulator) a pseudogap non-Fermi-liquid metal, a superconductor, and a normal metal appear, in striking similarity with the physics of cuprates. The strongly correlated s-wave superconducting state has a larger Drude weight than the corresponding normal state. The role of the impurity Kondo problem is underscored.
Recently developed numerical methods have enabled the explicit construction of the superconducting state of the Hubbard model of strongly correlated electrons in parameter regimes where the model also exhibits a pseudogap and a Mott insulating phase. d(x(2)-y(2)) symmetry superconductivity is found to occur in proximity to the Mott insulator, but separated from it by a pseudogapped nonsuperconducting phase. The superconducting transition temperature and order parameter amplitude are found to be maximal at the onset of the normal-state pseudogap. The emergence of superconductivity from the normal state pseudogap leads to a decrease in the excitation gap. All of these features are consistent with the observed behavior of the copper-oxide superconductors.
An intricate interplay between superconductivity, pseudogap, and Mott transition, either bandwidth driven or doping driven, occurs in materials. Layered organic conductors and cuprates offer two prime examples. We provide a unified perspective of this interplay in the two-dimensional Hubbard model within cellular dynamical mean-field theory on a 2×2 plaquette and using the continuous-time quantum Monte Carlo method as impurity solver. Both at half filling and at finite doping, the metallic normal state close to the Mott insulator is unstable to d-wave superconductivity. Superconductivity can destroy the first-order transition that separates the pseudogap phase from the overdoped metal, yet that normal state transition leaves its marks on the dynamic properties of the superconducting phase. For example, as a function of doping one finds a rapid change in the particle-hole asymmetry of the superconducting density of states. In the doped Mott insulator, the dynamical mean-field superconducting transition temperature T(c)(d) does not scale with the order parameter when there is a normal-state pseudogap. T(c)(d) corresponds to the local pair formation temperature observed in tunneling experiments and is distinct from the pseudogap temperature.
A recently proposed configuration-interaction-based impurity solver is used in combination with the single-site and four-site cluster dynamical mean field approximations to investigate the three-band copper oxide model believed to describe the electronic structure of high transition temperature copper-oxide superconductors. Use of the configuration interaction solver enables verification of the convergence of results with respect to the number of bath orbitals. The spatial correlations included in the cluster approximation substantially shift the metal-insulator phase boundary relative to the prediction of the single-site approximation and increase the predicted energy gap of the insulating phase by about 1 eV above the single-site result. Vertex corrections occurring in the four-site approximation act to dramatically increase the value of the optical conductivity near the gap edge, resulting in better agreement with the data. The calculations reveal two distinct correlated insulating states: the "magnetically correlated insulator," in which nontrivial intersite correlations play an essential role in stabilizing the insulating state, and the strongly correlated insulator, in which local physics suffices. Comparison of the calculations to the data places the cuprates in the magnetically correlated Mott insulator regime.
For doped two-dimensional Mott insulators in their normal state, the challenge is to understand the evolution from a conventional metal at high doping to a strongly correlated metal near the Mott insulator at zero doping. To this end, we solve the cellular dynamical mean-field equations for the two-dimensional Hubbard model using a plaquette as the reference quantum impurity model and continuous-time quantum Monte Carlo method as impurity solver. The normal-state phase diagram as a function of interaction strength UUU, temperature TTT, and filling nnn shows that, upon increasing nnn toward the Mott insulator, there is a surface of first-order transition between two metals at nonzero doping. That surface ends at a finite temperature critical line originating at the half-filled Mott critical point. Associated with this transition, there is a maximum in scattering rate as well as thermodynamic signatures. These findings suggest a new scenario for the normal-state phase diagram of the high temperature superconductors. The criticality surmised in these systems can originate not from a T === 0 quantum critical point, nor from the proximity of a long-range ordered phase, but from a low temperature transition between two types of metals at finite doping. The influence of Mott physics therefore extends well beyond half-filling.