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

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, one Mott-like insulating (M-phase) and one proximitized superconducting phase (S-phase), separated by first-order transition with hysteresis. In the M-phase the large correlation 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.

Figures (13)

• Figure 1: (a,b) Schematic representations of surface and Josephson junction problems. $\Delta$ and $\phi$ are the superconducting gap and phase, $\Gamma$ are the tunneling hybridization strengths between correlated (gray) and superconducting regions (orange). (c) Phase diagram at half filling and zero temperature with superconducting (S) and Mott (M) phases. $U_{c1}$ and $U_{c2}$ are spinodal interaction strengths that bound the coexistence region. $I_c$ denotes the critical current.
• Figure 1: Normal (diagonal component in Nambu space) local spectral function, $\mathcal{A}_{11}$, and anomalous (off-diagonal component in Nambu space) local spectral function, $\mathcal{A}_{12}$, as well as the corresponding imaginary parts of the self-energy, $\Sigma"_{11}$ and $\Sigma"_{12}$, for $U=2.6$, $\Delta=0.05$ and $\phi=0$ at various $\Gamma$. Left column: results starting from an insulating initial guess. Right column: results starting from a metallic initial guess. Note that $U$ belongs to the coexistence region, $U_{c1} < U < U_{c2}$.
• Figure 2: Normal (diagonal) spectral function $\mathcal{A}_{11}(\omega)$ on linear (a) and log scale (b), and anomalous (off-diagonal) spectral function $\mathcal{A}_{12}(\omega)$ on linear (d) and log scale (e) at $U=3.2 > U_{c2}(\Gamma=0)$ for a range of $\Gamma$. The transition occurs for $\Gamma_c=0.13$. (c, f) Imaginary parts of the (c) normal and (f) anomalous components of the self-energy. Insets: (a) zoom into the low-frequency region, (b) induced gap $\Delta^{\ast}$ vs. $\Gamma$ for a range of $U > U_{c1}$ and (c) the position of the $\delta$ peak in the normal part of the self-energy vs. $\Gamma$.
• Figure 2: Spin and charge susceptibility for $U = 2.4$ (left column) and $U = 3.1$ (right column) for several values of $\Gamma$. For reference we also plot the spectral function $\mathcal{A}_{11}$ (dashed lines).
• Figure 3: (a) Local susceptibilities for $\Gamma=0.05$ and (c) $\Gamma=0.2$ with $U=3.2$. We also show the diagonal component of the spectral function $\mathcal{A}_{11}$ for reference. (b) Local pair expectation value $\tau = U \langle d^{\dagger}_{\uparrow} d^{\dagger}_{\downarrow}\rangle$ and (d) double occupancy $U \langle n_{\uparrow} n_{\downarrow} \rangle$ vs. $\Gamma$.