Andreev bound state spectroscopy of a quantum-dot-based Aharonov-Bohm interferometer with superconducting terminals

Abstract

We analytically and numerically investigate an Aharonov-Bohm interferometer with two superconducting terminals and a strongly correlated quantum dot in one arm. Through a rigorous derivation, we prove that this double-path interferometer is spectrally equivalent to a simpler system: an interacting quantum dot coupled to a non-interacting side-coupled proximitized mode and a semiconductor lead. This equivalence reveals a simple interpretation of the interferometer's behavior through the competition of a geometric factor $χ$, a key parameter characterizing the anomalous part of the hybridization function, with the properties of the side-coupled mode. We identify the conditions for the formation of doublet chimney in the phase diagrams in more general setting. Moreover, we show how the obtained Andreev bound state spectra clearly indicate the presence of Josephson diode effect generated by interferometric phenomena.

Figures (9)

• Figure 1: $(a)$ Two-terminal interferometric device forming an Aharonov-Bohm ring with a direct electron hopping $t$ with phase $\Phi_t$ in one arm and a quantum dot (gray) in the other arm. The phases of electron hopping onto the dot are $\Phi_{V,1}$ and $\Phi_{V,2}$. Red arrows show the hopping phase orientations. Magnetic flux $\Phi_B$ pierces the ring, while $\Phi_{\varphi}$ pierces the superconducting loop (green). $(b)$ General $n$-terminal superconducting Anderson impurity model with the direct hopping $t$ (red) between the superconducting leads (green) and a centrally-placed strongly-interacting dot (gray).
• Figure 2: $(a)$ Schematics of a two-terminal SC-AIM with BCS leads (green) and a strongly-interacting single level QD (gray). Magnetic flux $\Phi_{\varphi}$ controls the phase difference $\varphi \equiv \varphi_2-\varphi_1 + \phi_{V,2} - \phi_{V,1}$. $(b)$ Phase-dependence of the energy $E$ of the sub-gap states for $U/\Delta=3$, $\Gamma/\Delta=1$, $\varepsilon_d=-U/2$. $(c)$ At $\varphi=\pi$, singlet (letter S) and doublet (D) ground states form a distinct pattern of a doublet chimney when plotted as $\nu \equiv 1/2-\varepsilon_d/U$ vs $V \equiv V_1 = V_2$ (data according to the minimal model of Pavesic-2024 with $U/\Delta = 10$ and $t/\Delta = 0.2$). $(d)$ Hybridization functions $D^W(\omega)$ at $\varphi/\pi=0, 0.75, 1.00$ in the insulator basis of Ref. Zalom-2021.
• Figure 3: $(a)$ Subsection \ref{['subsec:eom']} devises the EOM technique for the interferometric problem. $(b)$\ref{['subsec:self_energy_2']} demonstrates that the resulting self-energy $\mathbb{\Sigma}(z)$ exhibits diverse singularities in the complex plane. $(c)$\ref{['subsec:chi']} defines geometric factor $\bm{\chi}$ consisting of four complex-valued contributions $\gamma_1 e^{i (\varphi_1+\phi_{V,1})}$, $\gamma_1 \tilde{t}^2 e^{i \left(\varphi_2 + \phi_{V,1} + \phi_t \right)}$, $\gamma_2 e^{i (\varphi_2 + \phi_{V,2})}$ and $\gamma_2 \tilde{t}^2 e^{i \left(\varphi_1 + \phi_{V,2} - \phi_t \right) }$, which can be rotated counter-clockwise via $\varphi_j \rightarrow \varphi_j-\delta$ due to the gauge freedom. Consequently, real-valued $\bm{\chi}$ can be selected. $(d)$ In \ref{['subsec:side_mode']}, the isolated poles of self-energy are reinterpreted as a side-coupled mode of the interacting QD (blue). $(e)$ In \ref{['subsec:bogoliubov_valatin']} we employ real-valued $\bm{\chi}$ along with the Bogolyubov-Valatin transformations to map the original problem with two SC terminals (green) onto a one-terminal isolator (magenta) problem. $(f)$ Sec. \ref{['sec:nrg']} implements the log-gap NRG to $(e)$ by attaching original QD (orange) to one mode (blue) and a Wilson chain of non-interacting QDs (white).
• Figure 4: (a1)-(c1) Pole position $\omega_{\mathrm{pole}}$, $s^\mathrm{sv} (\omega_{\mathrm{pole}})$ and the geometric factor $\chi$ determine completely the properties of the side-coupled mode. The $\omega_{\mathrm{pole}}$ vs. $\varphi_t$ curves are identical except for the $\varphi$-dependent offsets. $s^{\mathrm{sv}}(\omega_{\mathrm{pole}})$ and $\chi$ contain harmonic functions of various linear combinations of $\varphi$ and $\varphi_t$. Consequently, they change their shape as well as root positions (blue and red circles). (a2)-(c2) $\Gamma_p$ depends only on model constants and $\omega_p$, thus it shifts correspondingly but does not change its shape. To accommodate for the condition $\varepsilon_p^2+\Delta_p^2=\omega_{\mathrm{pole}}^2$ both the energy of the side-coupled mode $\varepsilon_p$ as well as the induced pairing $\Delta_p$ change shape and shift the positions of the roots (blue and red circles). At $\varphi_t=-\varphi$ the roots of $s^{\mathrm{sv}}(\omega_{\mathrm{pole}})$ and $\chi$ align at the point where $\omega_{\mathrm{pole}}=\Delta$ and the mode decouples due to $\Gamma_p=0$. When additionally $\chi=0$, the AB ring obtains an additional symmetry. This manifests in the $W$ basis as a particle-hole symmetric hybridization function $D^W_{\mathrm{const}}$. In general, $D^W_{\mathrm{const}}$ is particle-hole asymmetric, as shown in panels (a3)-(c3). The particle-hole symmetric $D^W_{\mathrm{const}}$ guarantees a doublet GS which serves as a center of the doublet chimney as in the ordinary SC-AIM shown in Figs. \ref{['fig:sciam']}(d1)-(d3). Other parameter values are $\Gamma_1/\Delta = 0.25$, $\Gamma_2/\Delta = 0.25$, $\varphi=\pi/2$ and $\tilde{t}/\Delta = 1.5$.
• Figure 5: Many-body spectra of the DQD system comprised of the interacting QD and the side-coupled mode $p$ for three values of $U$. The model parameters are the same as in Fig. \ref{['fig:dqd']} with $\varphi=\pi$. The interacting dot is kept at half-filling, $\varepsilon_d=-U/2$. The vertical gray line denotes the decoupling of the DQD system for $\Gamma_p=0$ which guarantees an overall doublet GS. With varying $\varphi$ the spectra only shift as the decoupling condition moves according to $\varphi_t=-\varphi$.