Double-quantum-dot Andreev molecules: Phase diagrams and critical evaluation of effective models

Abstract

This work systematically investigates the phase diagram of a parallel double-quantum-dot Andreev molecule, where the two quantum dots are coupled to a common superconducting lead. Using the numerical renormalization group method, we map out the evolution of the ground state across a wide parameter space of level detunings, size of the superconducting gap, lead couplings, and inter-dot coupling strength. The intricate phase diagrams feature singlet, doublet, and a relatively uncommon triplet ground states, with the latter being a distinct signature of strong lead-mediated interactions between the quantum dots. We benchmark the applicability of simplified effective models, including the atomic limit and zero-bandwidth approximations, in capturing the complex behavior of this parallel configuration. Our analysis reveals severe limitations of these models, underscoring the necessity for maximal caution when extrapolating beyond their tested validity. In particular, all effective models except for the extended version of the zero-bandwidth approximation failed in reproducing the triplet ground state and made several false predictions. These findings provide crucial insights for interpreting experimental observations and designing superconducting devices based on quantum-dot architectures.

Figures (11)

• Figure 1: (a) Schematic of the considered system, consisting of two QDs (gray circles) coupled to a single BCS channel, with couplings $\Gamma_1$ and $\Gamma_2$ and hopping between the dots denoted by $t$. Such scenario is achieved when the distance between QDs is much smaller than the superconducting coherence length $\zeta$. In the other limit, shown in (b), the DQD system can be well understood as a two channel model. (c) When $\Delta=0$ and $\Gamma_1, \Gamma_2, t \rightarrow 0$ the presence of only one screening channel leads to the formation of a doublet GS (yellow) in the middle of the phase diagram due to the underscreened Kondo effect. Changing the occupation of the dots by one electron leads then to the formation of singlet GS (white). The corresponding QD occupations $(n_1, n_2)$ are indicated. (d) For two screening channels, when $\Delta=0$, full screening of the DQD system is possible with the phase diagram consisting only of a singlet GS when the $\Gamma_1, \Gamma_2, t \rightarrow 0$. (e)-(f) For $\Delta$ being the dominant energy scale, no Kondo screening is possible and a checkerboard pattern of singlet and doublet GS emerges for one as well as two screening channels. Note that the location of the voided crossings (red circles) is distinctly different in both cases. Notice also that we show the phase diagrams as functions of negative detunings, which directly corresponds to experimental gate voltage sweeps.
• Figure 2: The NRG results for the phase diagrams for the DQD system of two identical QDs ($U_1=U_2\equiv U$), as functions of the detuning parameters $-\delta_1/U$ and $-\delta_2/U$ for different values of the superconducting energy gap $\Delta/U$, as indicated. Coupling-symmetric scenario $\Gamma_1/U=\Gamma_2/U\equiv \Gamma/U=0.05$ maximizing lead-mediated correlations ($\nu=1$) is selected. The white, yellow, and green regions correspond to singlet ($S=0$), doublet ($S=1/2$), and triplet ($S=1$) GS, respectively. (a) In the absence of superconductivity ($\Delta=0$), non-zero cross-correlations mediated by the common lead result in a doublet GS island at the center of the phase diagram. (b)-(c) As $\Delta/U$ increases from $5\times10^{-5}$ to $10^{-4}$, the doublet GS island elongates into a cross-shaped region. (d)-(e) For $\Delta/U=5\times10^{-4}$, a triplet GS emerges in the center of the phase diagram. (f)-(g) The cross-shaped doublet region splits into two separate stripes along the main diagonal $\delta_1=\delta_2$, and the triplet region begins to shrink. (h) As $\Delta/U$ increases further to $0.5$, the triplet GS disappears, and a striped pattern of singlet and doublet phases appears in the direction of the main diagonal $\delta_1=\delta_2$. All calculations have been performed via NRG with discretization parameter $\Lambda=2$ exploiting full spin symmetry with at least $1000$ multiplets kept during calculations, assuming $U=0.2D$ and the band halfwidth $D \equiv 1$ used as energy unit.
• Figure 3: The NRG results for the phase diagrams for the coupling symmetric system of two identical parallel QDs at $\nu=1$ as a function of the detuning parameters $-\delta_1/U$ and $-\delta_2/U$ for varying superconducting gap $\Delta/U$. The white, yellow, and green regions correspond to the singlet ($S=0$), doublet ($S=1/2$), and triplet ($S=1$) GS, respectively. (a1)-(a5) For $\Gamma/U=0.1$, the evolution with increasing $\Delta/U$ is qualitatively similar to Fig. \ref{['fig:nrg_UG_20']}, where a cross-shaped doublet region splits into separate stripes, and a triplet GS emerges and then disappears as $\Delta/U$ increases. (b1)-(b5) For $\Gamma/U=0.2$, the Kondo screening effects dominate over the lead-mediated correlations, preventing the formation of a triplet GS for any value of $\Delta/U$. (c1)-(c5) At a stronger coupling of $\Gamma/U=0.5$, the Kondo screening processes dominate the behavior, resulting in phase diagrams with only centrally-positioned doublet region immersed in a singlet GS area. The other parameters are the same as in Fig. \ref{['fig:nrg_UG_20']}.
• Figure 4: The NRG results for the phase diagrams for the coupling symmetric system of two identical parallel QDs at $\nu=1$ calculated as a function of the detuning parameters $-\delta_1/U$ and $-\delta_2/U$ for varying inter-dot hopping strength $t/U$ for $\Gamma /U = 0.05$. The white, yellow, and green regions correspond to singlet (S=0), doublet (S=1/2), and triplet (S=1) GS, respectively. The rows correspond to different values of the superconducting energy gap (a) $\Delta/U = 5\times10^{-5}$, (b) $5\times10^{-4}$, (c) $5\times10^{-3}$ and (d) $5\times10^{-2}$. (a1)-(a5) Initially, with the interdot hopping set to $t/U=0$, the phase diagrams exhibit a doublet GS island, which gets distorted as $t/U$ increases. (b1)-(b5), (c1)-(c5), (d1)-(d5) For larger values of $\Delta/U$, a triplet GS emerges initially that shrinks and disappears entirely, as $t/U$ is increased. Finally, one stripe of the doublet phase remains immersed in the singlet GS region. For$t/U \neq 0$, the emerging asymmetry across the diagonal $\delta_1=-\delta_2$, is governed by the ph-transformation $\mathcal{T}_1$ discussed in Sec. \ref{['sec:app_ph']}. Except of $t/U$, all parameters are the same as in Fig. \ref{['fig:nrg_UG_20']}.
• Figure 5: Phase diagrams of two identical parallel dots ($U_1 = U_2 \equiv U$) calculated using the AL theory for the symmetric coupling scenario with $\Gamma^{\mathrm{AL}}_1 = \Gamma^{\mathrm{AL}}_2 \equiv \Gamma^{\mathrm{AL}}$ at $\nu=1$ as a function of the detuning parameters, $-\delta_1/U$ and $-\delta_2/U$. At $t=0$, $\Gamma^{\mathrm{AL}}/U$ remains the only relevant scale to be varied in the AL theory. Panels (a)-(c) with $\Gamma^{\mathrm{AL}}/U \leq 0.25$ are in a good qualitative agreement with the NRG results observed for large $\Delta$ in Figs. \ref{['fig:nrg_UG_20']} and \ref{['fig:nrg_UG']}. However, the remaining panels (d)-(f), with larger $\Gamma^{\mathrm{AL}}/U$ values, have no counterpart in the exact results. Overall, the AL fails in predicting the triplet GS completely. The same also applies to GAL, as it only rescales the parameters that enter the AL theory.