Strongly correlated electrons in superconducting islands with fluctuating Cooper pairs

We present a particle-number-conserving theory for many-body effects in mesoscopic superconducting islands connected to normal electrodes, which explicitly includes quantum fluctuations of Cooper pairs in the condensate. Beyond previous BCS mean-field descriptions, our theory can precisely treat the pairing and Coulomb interactions over a broad range of parameters by using the numerical renormalization group method. On increasing the ratio of pairing interactions to Coulomb interactions, the low-energy physics of the system evolves from the spin Kondo regime to the mixed-valence regime and eventually reaches an anisotropic charge Kondo phase, while a crossover from $1e$- to $2e$-periodic Coulomb blockade of transport is revealed at high temperatures. For weak pairing, the superconducting condensate is frozen in the local spin-flip processes but fluctuates in the virtual excitations, yielding an enhanced spin Kondo temperature. For strong pairing, massive fluctuations of Cooper pairs are crucial for establishing charge Kondo correlations whose Kondo temperature rapidly decreases with the pairing interaction. Surprisingly, a charge-exchange-induced local field may occur even at the charge degenerate point, thereby destroying the charge Kondo effect. These are demonstrated in the spectral and transport properties of the island.

Figure 1

Eigenenergy $E_N^{\sigma ,\nu }$ of isolated SC QDs as a function of gate voltage $V_g$ for different numbers $N$ of dot electrons, in the regimes of weak [(a) $\mathrm{\Delta }/{\mathrm{\Delta }}_C=0.2$] and strong [(b) $\mathrm{\Delta }/{\mathrm{\Delta }}_C=2$] SC correlations. Numbers in (a) indicate the value of $N-2N_0$ of each parabola. Here, the $d$ level is set at ${\varepsilon }_0=0$.

Figure 2

Zero-temperature spectral properties of $d$ electrons in SC islands. (a) Spectral density $A\left(\varepsilon \right)$ vs positive energy $\varepsilon >0$, for different SC pairings, $\mathrm{\Delta }/{\mathrm{\Delta }}_C=0$ (blue), 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 (red). (b) Same as (a) but for $\mathrm{\Delta }/{\mathrm{\Delta }}_C=1.1$ (blue), 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, and 2.0 (red). Inset: $A\left(\varepsilon \right)$ in the full range of $\varepsilon$. (c) Half-width at half maximum $T_0$ of the central peak in $A\left(\varepsilon \right)$ as a function of $\mathrm{\Delta }$. The two solid lines are fits by two analytical expressions of Kondo temperatures, with $A_c=0.02686$ and $A_s=0.02741$. (d) Suppression of the charge Kondo resonance by the $d$-level ${\varepsilon }_0$ (${\delta }_1=2{\varepsilon }_0$) and the gate voltage $V_g [{\delta }_2=2E_C(2N_0+1-V_g)]$ deviating from the symmetric point.

Figure 3

(a)–(f) Conductance $G$ as a function of the gate voltage $V_g$ at zero temperature $T=0$ [(a)–(c)] and high temperature $T=2\mathrm{\Gamma }$ [(d)–(f)], for several values of the SC pairing $\mathrm{\Delta }$. Inset of (c): Zoom into the resonance at $V_g=2N_0+1$. (g) and (h) Conductance as a function of the temperature $T$ scaled by $\mathrm{\Gamma }$ and $T_0$, respectively, for different $\mathrm{\Delta }$. The vertical dotted line in (g) indicates the position of $T=2\mathrm{\Gamma }$. In (h), two blue vertical dotted lines mark the positions of $T=2\mathrm{\Gamma }$ (left) and $E^{*}$ (right) for $\mathrm{\Delta }/{\mathrm{\Delta }}_C=1.5$, with $E^{*}$ the excitation energy (see text). Two red vertical dotted lines represent the same but for $\mathrm{\Delta }/{\mathrm{\Delta }}_C=1.8$.