Nonlocal Andreev transport through a quantum dot in a magnetic field:
Interplay between Kondo, Zeeman, and Cooper-pair correlations

We start with an Anderson impurity model for a single quantum dot (QD) connected to two normal (N) and one superconducting (SC) leads, as shown in Fig. 1:

	$\displaystyle H\,=$	$\displaystyle\ \,H_{\mathrm{dot}}\,+\,H_{\text{N}}\,+\,H_{\text{TN}}\,+\,H_{\text{S}}+H_{\text{TS}}\,,$		(1)
	$\displaystyle H_{\mathrm{dot}}=$	$\displaystyle\ \xi_{d}\bigl{(}n_{d}-1\bigr{)}-b\,\bigl{(}n_{d,\uparrow}-n_{d,\downarrow}\bigr{)}\,+\frac{U}{2}\bigl{(}n_{d}-1\bigr{)}^{2},$		(2)
	$\displaystyle H_{\text{N}}=$	$\displaystyle\ \sum_{\nu=L,R}\sum_{\sigma}\int_{-D}^{D}\!d\varepsilon\,\varepsilon\,c^{\dagger}_{\varepsilon,\nu,\sigma}c_{\varepsilon,\nu,\sigma},$		(3)
	$\displaystyle H_{\text{TN}}=$	$\displaystyle\ \sum_{\nu=L,R}v_{\nu}\sum_{\sigma}\int_{-D}^{D}\!d\varepsilon\,\sqrt{\rho_{c}}\,\Bigl{(}c^{\dagger}_{\varepsilon,\nu,\sigma}d_{\sigma}+\mathrm{H.c.}\Bigr{)},$		(4)
	$\displaystyle H_{\text{S}}=$	$\displaystyle\ \sum_{\sigma}\!\int_{-D_{S}}^{D_{S}}\!\!d\varepsilon\,\varepsilon\,s^{\dagger}_{\varepsilon,\sigma}s_{\varepsilon,\sigma}$
		$\displaystyle\quad\ +\int_{-D_{S}}^{D_{S}}\!\!d\varepsilon\left(\Delta_{S}\,s^{\dagger}_{\varepsilon,\uparrow}s^{\dagger}_{\varepsilon,\downarrow}+\mathrm{H.c.}\right),$		(5)
	$\displaystyle H_{\text{TS}}=$	$\displaystyle\ v_{\text{S}}\sum_{\sigma}\!\int_{-D_{S}}^{D_{S}}\!\!d\varepsilon\,\sqrt{\rho_{S}}\,\Bigl{(}s^{\dagger}_{\varepsilon,\sigma}d_{\sigma}+\mathrm{H.c.}\Bigr{)}.$		(6)

Here, $H_{\mathrm{dot}}$ describes the QD part: $\xi_{d}\equiv\varepsilon_{d}+{U}/{2}$, with $\varepsilon_{d}$ the discrete energy level, $U$ the Coulomb interaction, and $b$ ($\equiv\mu_{B}B)$ the Zeeman energy due to the magnetic field $B$ applied to the QD, with $\mu_{B}$ the Bohr magneton. $d^{\dagger}_{\sigma}$ is the creation operator for an electron with spin $\sigma$, and $n_{d}\equiv n_{d,\uparrow}+n_{d,\downarrow}$ is the number operator with $n_{d,\sigma}\equiv d^{\dagger}_{\sigma}d_{\sigma}$. A constant energy shift, which does not affect the physics, is included in Eq. (2) in order to describe clearly that the system has the electron-hole symmetry at $\xi_{d}=0$.

$H_{\text{N}}$ describes the conduction electrons in the normal leads, the density of states of which is assumed to be flat $\rho_{c}=1/(2D)$, with $D$ the half-width of the bands. $c^{\dagger}_{\varepsilon,\nu,\sigma}$ is the creation operator for conduction electrons with spin $\sigma$ and energy $\varepsilon$. The operators for conduction electrons satisfy the following anti-commutation relation that is normalized by the Dirac delta function: $\{c_{\varepsilon,\nu,\sigma},c^{\dagger}_{\varepsilon^{\prime},\nu^{\prime},\sigma^{\prime}}\}=\delta_{\nu\nu^{\prime}}\,\delta_{\sigma\sigma^{\prime}}\delta(\varepsilon-\varepsilon^{\prime})$. $H_{\text{TN}}$ describes the tunnel couplings between the QD and the normal leads. The level broadening of the discrete energy level in the QD is given by $\Gamma_{N}\equiv\Gamma_{L}+\Gamma_{R}$, with $\Gamma_{\nu}\equiv\pi\rho_{c}v_{\nu}^{2}$ the contributions of the two normal leads on the left $\nu=L$ and right $\nu=R$.

$H_{\text{S}}$ and $H_{\text{TS}}$ describe the contributions of the superconducting lead with an $s$-wave SC gap $\Delta_{S}\equiv\left|\Delta_{S}\right|\,e^{i\phi_{S}}$: $s^{\dagger}_{\varepsilon,\sigma}$ is the creation operator for electrons in the SC lead, with $D_{S}$ the half-band width and $\rho_{S}=1/(2D_{S})$. One of the key parameters for the SC proximity effects is $\Gamma_{S}\equiv\pi\rho_{S}v_{\text{S}}^{2}$, i.e., the coupling strength between the QD and the SC lead.

In this paper, we study the crossed Andreev reflection occurring at low energies, much lower than the SC energy gap. To this end, we consider the large gap limit $\left|\Delta_{S}\right|\to\infty$, which is taken at $\left|\Delta_{S}\right|\ll D_{S}$ keeping $\Gamma_{S}$ constant.Tanaka et al. (2007a) In this case, the superconducting proximity effects can be described by the pair potential penetrating into the QD:

$\displaystyle\Delta_{d}\equiv\Gamma_{S}\,e^{i\phi_{S}}\,.$

(7)

The Coulomb interaction $U$ induces the correlation effects for electrons in the QD and the symmetrized linear combination of the conduction bands defined in Eq.  (46), which can be described by an effective Hamiltonian $H_{\mathrm{eff}}$ given in Eq. (50) (see Appendix A). Furthermore, carrying out the Bogoliubov rotation defined in Eq. (53), it can be transformed further into a system of interacting Bogoliubov particles described by a standard Anderson model:

	$\displaystyle H_{\mathrm{eff}}=$	$\displaystyle\ E_{A}\left(\sum_{\sigma}\gamma^{\dagger}_{d,\sigma}\gamma_{d,\sigma}-1\right)-b\,\Bigl{(}\gamma^{\dagger}_{d,\uparrow}\gamma_{d,\uparrow}-\gamma^{\dagger}_{d,\downarrow}\gamma_{d,\downarrow}\Bigr{)}$
		$\displaystyle+\frac{U}{2}\left(\sum_{\sigma}\gamma^{\dagger}_{d,\sigma}\gamma_{d,\sigma}-1\right)^{2}+\sum_{\sigma}\int_{-D}^{D}\!d\varepsilon\,\varepsilon\ \gamma^{\dagger}_{\varepsilon,\sigma}\gamma_{\varepsilon,\sigma}$
		$\displaystyle+v_{N}\sum_{\sigma}\int_{-D}^{D}\!d\varepsilon\,\sqrt{\rho_{c}}\,\left(\gamma^{\dagger}_{\varepsilon,\sigma}\gamma_{d,\sigma}+\mathrm{H.c.}\right)\,,$		(8)
	$\displaystyle N_{\gamma}=$	$\displaystyle\ \sum_{\sigma}\gamma^{\dagger}_{d,\sigma}\gamma_{d,\sigma}+\sum_{\sigma}\int_{-D}^{D}\!d\varepsilon\,\gamma^{\dagger}_{\varepsilon,\sigma}\gamma_{\varepsilon,\sigma}\,.$		(9)

Here, $E_{A}\equiv\sqrt{\xi_{d}^{2}+\Gamma_{S}^{2}}$ is the effective impurity level, and $v_{N}\equiv\sqrt{v_{L}^{2}+v_{R}^{2}}$. The operators $\gamma_{d,\sigma}$ and $\gamma_{\varepsilon,\sigma}$ describe the Bogoliubov particles in the dot and the symmetrized part of the conduction band, respectively. The effective Hamiltonian conserves the total number of the Bogoliubov particles $N_{\gamma}$, reflecting the $U(1)$ symmetry along the principal axis in the Nambu pseudo-spin space.

Figure 2 illustrates the parameter space of $H_{\mathrm{eff}}$ at zero magnetic field $b=0$. For finite $\Gamma_{N}$, the Kondo screening due to the normal conduction electrons occurs inside the semicircle region, at which the impurity level is occupied by a single Bogoliubov particle: $Q\simeq 1.0$ with

$\displaystyle Q\,$

$\displaystyle\equiv\,Q_{\uparrow}+Q_{\downarrow},\qquad Q_{\sigma}\,\equiv\,\left\langle\gamma^{\dagger}_{d,\sigma}\gamma_{d,\sigma}\right\rangle.$

(10)

Bogoliubov particles show also the valence-fluctuation behavior near $E_{A}\simeq U/2$, at which the crossover between the Kondo singlet and the superconducting singlet occurs. The Bogoliubov rotation angle corresponds to $\Theta=\cot^{-1}(\xi_{d}/\Gamma_{S})$ shown in Fig. 2. In particular, the crossed Andreev scattering is enhanced in the angular range of $\pi/4\lesssim\Theta\lesssim 3\pi/4$ outside the semicircle $E_{A}\gtrsim U/2$, as discussed later in Secs. IV and V.

In this work, we calculate the nonlocal conductance for the current flowing into the drain electrode, using the retarded Green’s function for electrons in the QD:

	$\displaystyle\!\!\!\bm{G}^{r}_{dd}(\omega)\equiv-i\int_{0}^{\infty}\!dt\,e^{i\left(\omega+i0^{+}\right)t}$
	$\displaystyle\qquad\qquad\ \ \times\begin{pmatrix}\left\langle\left\{d_{\uparrow}(t),\,d^{\dagger}_{\uparrow}\right\}\right\rangle&\left\langle\left\{d_{\uparrow}(t),\,d_{\downarrow}\right\}\right\rangle\\ \left\langle\left\{d^{\dagger}_{\downarrow}(t),\,d^{\dagger}_{\uparrow}\right\}\right\rangle&\left\langle\left\{d^{\dagger}_{\downarrow}(t),\,d_{\downarrow}\right\}\right\rangle\rule{0.0pt}{17.07182pt}\end{pmatrix}.$		(11)

Here, $\langle\cdots\rangle$ denotes the thermal average at equilibrium. This matrix Green’s function can be diagonalized with the Bogoliubov transformation:

$\displaystyle\bm{\mathcal{U}}^{\dagger}\,\bm{G}^{r}_{dd}(\omega)\ \bm{\mathcal{U}}\,$

$\displaystyle=\,\begin{pmatrix}G^{r}_{\gamma,\uparrow}(\omega)&0\\ 0&-G^{a}_{\gamma,\downarrow}(-\omega)\rule{0.0pt}{11.38092pt}\end{pmatrix}.$

(12)

We will choose the Josephson phase of the pair potential to be $\phi_{S}=0$ in the following, so that $\bm{\mathcal{U}}$ is determined solely by a pseudo-spinor rotation with the angle $\Theta/2$:

$\displaystyle\bm{\mathcal{U}}\,$

$\displaystyle=\,\begin{pmatrix}\cos\frac{\Theta}{2}&\,-\sin\frac{\Theta}{2}\\ \sin\frac{\Theta}{2}&\rule{0.0pt}{14.22636pt}\quad\cos\frac{\Theta}{2}\end{pmatrix}.$

(13)

The matrix elements of $\bm{\mathcal{U}}$ determine the behaviors of transport coefficients as the superconducting coherence factors,

$\displaystyle\cos\frac{\Theta}{2}=\sqrt{\frac{1}{2}\left(1+\frac{\xi_{d}}{E_{A}}\right)},\quad\sin\frac{\Theta}{2}=\sqrt{\frac{1}{2}\left(1-\frac{\xi_{d}}{E_{A}}\right)}.\rule{0.0pt}{28.45274pt}$

The diagonal elements $G^{r}_{\gamma,\sigma}$ and $G^{a}_{\gamma,\sigma}$ on the right-hand side of Eq. (12) are the retarded and advanced Green’s functions for the interacting Bogoliubov particles, described by $H_{\mathrm{eff}}$. These diagonal elements can be expressed in the form, using Eq. (53),

	$\displaystyle G^{r}_{\gamma,\sigma}(\omega)\equiv$	$\displaystyle\,-i\int_{0}^{\infty}\!dt\,e^{i\left(\omega+i0^{+}\right)t}\left\langle\left\{\gamma_{d,\sigma}(t),\,\gamma^{\dagger}_{d,\sigma}\right\}\right\rangle$
	$\displaystyle=$	$\displaystyle\ \frac{1}{\omega\,-E_{A,\sigma}-\Sigma^{U}_{\gamma,\sigma}(\omega)+i\Gamma_{N}},$		(14)

and $G^{a}_{\gamma,\sigma}(\omega)=\left\{G^{r}_{\gamma,\sigma}(\omega)\right\}^{*}$. Here, $E_{A,\sigma}\equiv E_{A}-\sigma\,b$, and $\Sigma^{U}_{\gamma,\sigma}(\omega)$ represents the self-energy corrections due to the Coulomb interaction term, $({U}/{2})\left(n_{d}-1\right)^{2}$, defined in Eq. (2). The unperturbed part of the denominator describes the Andreev resonance level with the width $\Gamma_{N}$ situated at $\omega=E_{A,\sigma}$.

At low energies, effects of the electron correlations on the transport properties can be deduced from the behavior of the self-energy near $\omega\simeq 0$ at zero temperature $T=0$:

$\displaystyle G^{r}_{\gamma,\sigma}(\omega)\,\simeq\,\frac{Z_{\sigma}}{\omega-\widetilde{E}_{A,\sigma}+i\widetilde{\Gamma}_{N,\sigma}}\,.$

(15)

The asymptotic form of the Green’s function defines a renormalized resonance level of quasiparticles in the Fermi liquid, the position $\widetilde{E}_{A,\sigma}$ and the width $\widetilde{\Gamma}_{N,\sigma}$ of which are given byNozières (1974); Yamada (1975a, b); Shiba (1975); Yoshimori (1976)

	$\displaystyle\widetilde{\Gamma}_{N,\sigma}=$	$\displaystyle\ Z_{\sigma}\,\Gamma_{N},\qquad\frac{1}{Z_{\sigma}}\,=\,1-\left.\frac{\partial\Sigma^{U}_{\gamma,\sigma}(\omega)}{\partial\omega}\right|_{\omega=0},$		(16)
	$\displaystyle\widetilde{E}_{A,\sigma}\,=$	$\displaystyle\ Z_{\sigma}\Bigl{[}\,E_{A,\sigma}+\Sigma^{U}_{\gamma,\sigma}(0)\,\Bigr{]}\,.$		(17)

Furthermore, the phase shift $\delta_{\sigma}$ of the interacting Bogoliubov particles is defined by $G^{r}_{\gamma,\sigma}(0)=-\left|G^{r}_{\gamma,\sigma}(0)\right|e^{i\delta_{\sigma}}$, i.e.,

$\displaystyle\delta_{\sigma}\,=$

$\displaystyle\ \frac{\pi}{2}-\tan^{-1}\left(\frac{\widetilde{E}_{A,\sigma}}{\widetilde{\Gamma}_{N,\sigma}}\right)\,,$

(18)

plays a primary role in the ground-state properties. These renormalized parameters can be calculated, for instance, using the NRG approach described in the next section.

The Friedel sum rule also holds for the interacting Bogoliubov particles, and thus the average number of the Bogoliubov particles in the QD is determined by the phase shift,

$\displaystyle Q_{\sigma}$

$\displaystyle\xrightarrow{\,T\to 0\,}\frac{\delta_{\sigma}}{\pi}.$

(19)

The phase shift varies in the range of $0\leq\delta_{\sigma}\leq\pi/2$ along the radial coordinate $E_{A}$ in the $\xi_{d}$ vs $\Gamma_{S}$ plane but is independent of the angle $\Theta$.

The ground-state properties, such as the occupation number of electrons $\left\langle n_{d}\right\rangle$ and the pair correlation function $\bigl{\langle}d^{\dagger}_{\uparrow}\,d^{\dagger}_{\downarrow}+d_{\downarrow}\,d_{\uparrow}\bigr{\rangle}$, can be deduced from $Q$:

	$\displaystyle\left\langle n_{d}\right\rangle-1\ =$	$\displaystyle\,\left(Q-1\right)\,\cos\Theta\,,$		(20)
	$\displaystyle\left\langle d^{\dagger}_{\uparrow}\,d^{\dagger}_{\downarrow}+d_{\downarrow}\,d_{\uparrow}\right\rangle\ =$	$\displaystyle\,\left(Q-1\right)\,\sin\Theta\,.$		(21)

These two averages correspond to the projection of a vector of magnitude $Q-1$ directed along the principal axis onto the $z$ axis and the $x$ axis of the Nambu space, respectively. Furthermore, a local magnetization $m_{d}$ is induced in the quantum dot at finite magnetic fields,

$\displaystyle m_{d}\,\equiv\$

$\displaystyle\ \left\langle n_{d,\uparrow}\right\rangle-\left\langle n_{d,\downarrow}\right\rangle\ =\,Q_{\uparrow}-Q_{\downarrow}\,.$

(22)

Therefore, the occupation number of electrons with spin $\sigma$ is given by

$\displaystyle\left\langle n_{d,\sigma}\right\rangle\,=\,\frac{1+\cos\Theta}{2}\,\frac{\delta_{\sigma}}{\pi}\,+\,\frac{1-\cos\Theta}{2}\,\left(1-\frac{\delta_{\overline{\sigma}}}{\pi}\right)\,,$

(23)

where $\overline{\sigma}$ represents an opposite-spin component of $\sigma$.

We consider the linear-response current $I_{R}$ flowing from the QD to the normal lead on the right, induced by bias voltages $V_{L}$ and $V_{R}$ applied to the left and right leads, respectively. It can be expressed in the following form at $T=0$ (see Appendix B):

	$\displaystyle I_{R}\,=$	$\displaystyle\ \,I_{R}^{\mathrm{ET}}\,+\,I_{R}^{\mathrm{CP}},$		(24)
	$\displaystyle I_{R}^{\mathrm{ET}}=$	$\displaystyle\ \frac{2e^{2}}{h}\mathcal{T}_{\mathrm{ET}}\ \frac{4\Gamma_{R}\Gamma_{L}}{\Gamma_{N}^{2}}\,\bigl{(}V_{L}-V_{R}\bigr{)},\rule{0.0pt}{17.07182pt}$
	$\displaystyle I_{R}^{\mathrm{CP}}=$	$\displaystyle\ -\frac{2e^{2}}{h}\mathcal{T}_{\mathrm{CP}}\left[\frac{4\Gamma_{R}\Gamma_{L}}{\Gamma_{N}^{2}}\,\bigl{(}V_{L}+V_{R}\bigr{)}+\frac{4\Gamma_{R}^{2}}{\Gamma_{N}^{2}}\,2V_{R}\right].$

Correspondingly, the current $I_{L}$ flowing from the left normal lead towards the QD takes the form, $I_{L}=I_{L}^{\mathrm{ET}}+I_{L}^{\mathrm{CP}}$, with $I_{L}^{\mathrm{ET}}=I_{R}^{\mathrm{ET}}$ and

$\displaystyle I_{L}^{\mathrm{CP}}=$

$\displaystyle\ \frac{2e^{2}}{h}\mathcal{T}_{\mathrm{CP}}\left[\frac{4\Gamma_{R}\Gamma_{L}}{\Gamma_{N}^{2}}\,\bigl{(}V_{L}+V_{R}\bigr{)}+\frac{4\Gamma_{L}^{2}}{\Gamma_{N}^{2}}\,2V_{L}\right].$

(25)

The two components of the current $I_{\nu}^{\mathrm{ET}}$ and $I_{\nu}^{\mathrm{CP}}$ represent the contribution of the single-electron tunneling and that of the Cooper-pair tunneling, respectively. The transmission probabilities $\mathcal{T}_{\mathrm{ET}}$ and $\mathcal{T}_{\mathrm{CP}}$ are determined by the equilibrium Green’s functions at the Fermi level $\omega=0$, and can be expressed in terms of the phase shifts and the Bogoliubov angle (see Appendix B):

	$\displaystyle\!\!\!\mathcal{T}_{\mathrm{ET}}\equiv$	$\displaystyle\ \frac{\Gamma_{N}^{2}}{2}\left[\ \Bigl{|}\bigl{\{}\bm{G}_{dd}^{r}(0)\bigl{\}}_{11}\Bigr{|}^{2}+\Bigl{|}\bigl{\{}\bm{G}_{dd}^{r}(0)\bigl{\}}_{22}\Bigr{|}^{2}\ \right]$
	$\displaystyle=$	$\displaystyle\ \frac{1}{2}\sum_{\sigma}\sin^{2}\delta_{\sigma}-\frac{1}{4}\,\sin^{2}\Theta\,\sin^{2}\bigl{(}\delta_{\uparrow}+\delta_{\downarrow})\,,$		(26)
	$\displaystyle\!\!\!\mathcal{T}_{\mathrm{CP}}\equiv$	$\displaystyle\ \frac{\Gamma_{N}^{2}}{2}\left[\ \Bigl{|}\bigl{\{}\bm{G}_{dd}^{r}(0)\bigl{\}}_{12}\Bigr{|}^{2}+\Bigl{|}\bigl{\{}\bm{G}_{dd}^{r}(0)\bigl{\}}_{21}\Bigr{|}^{2}\ \right]$
	$\displaystyle=$	$\displaystyle\,\frac{1}{4}\,\sin^{2}\Theta\,\sin^{2}\bigl{(}\delta_{\uparrow}+\delta_{\downarrow})\,.$		(27)

These two are bounded in the range, $0\leq\mathcal{T}_{\mathrm{ET}}\leq 1$ and $0\leq\mathcal{T}_{\mathrm{CP}}\leq 1/4$, and are related to each other through the optical theorem (see Appendix C):

$\displaystyle\mathcal{T}_{\mathrm{ET}}+\mathcal{T}_{\mathrm{CP}}\,=\,\mathcal{T}_{\mathrm{BG}}\,,\qquad\mathcal{T}_{\mathrm{BG}}\,\equiv\,\frac{1}{2}\,\sum_{\sigma}\sin^{2}\delta_{\sigma}\,.$

(28)

Here, $\mathcal{T}_{\mathrm{BG}}$ can be regarded as a transmission probability of the Bogoliubov particles.

The linear-response coefficients, given in the above for the large gap limit $|\Delta_{S}|\to\infty$, are determined by $\delta_{\sigma}$ and $\Theta$. Therefore, the Cooper-pair contributions, which vary depending on the parameter regions shown in Fig. 2, can systematically be explored by using the polar coordinate ($E_{A}$, $\Theta$) since the phase shift $\delta_{\sigma}$ through which the many-body effects enter is independent of the angle $\Theta$ that determines the superconducting coherence factor $\sin^{2}\Theta$ for the transmission probability $\mathcal{T}_{\mathrm{CP}}$.

In the following two sections, we numerically investigate the contribution of the CAR over a wide range of the parameter space. To this end, we have calculated the phase shift $\delta_{\sigma}$ and the other correlation functions of Bogoliubov quasiparticles, applying the NRG approachKrishna-murthy et al. (1980a, b); Hewson et al. (2004) to the effective Hamiltonian $H_{\mathrm{eff}}$ given in Eq. (8),Tanaka et al. (2007a) choosing the discretization parameter to be $\Lambda=2.0$ and $\Gamma_{N}/D=1/(100\pi)$. We have also constructed the interpolating functions for the phase shift $\delta_{\sigma}$ from a discrete set of the NRG data obtained along the radial-$E_{A}$ direction in the parameter space, described in Fig. 2. The dependence of the transport coefficients on the Bogoliubov-rotation angle $\Theta$ of the polar coordinate has been determined by using the exact formulas presented in the above.

We will discuss the CAR contribution to the nonlocal transport at zero field in Sec. IV, and then consider magnetic-field dependence in Sec. V.

We next consider how the ground state of $H_{\mathrm{eff}}$ evolves as $E_{A}$ varies along the radial direction in the $\xi_{d}$ vs $\Gamma_{S}$ plane, shown in Fig. 2. Note that the eigenstates and eigenvalues of the effective Hamiltonian defined in Eq. (8) do not depend on the angular coordinate $\Theta$.

Figure 3(a) shows the occupation number $Q$ as a function of $E_{A}$ for $U/(\pi\Gamma_{N})=1.0$, $2.0$, $3.0$, and $5.0$. We see that $Q$ decreases as $E_{A}$ increases, especially near $E_{A}\simeq U/2$, where the crossover from the Kondo regime to the valence-fluctuation regime of Bogoliubov particles occurs for large interactions $U/(\pi\Gamma_{N})\gtrsim 2.0$. Figure 3(b) shows the magnitude of the pair correlation function for $\Theta=\pi/2$, where the absolute value is given by $|\bigl{\langle}d^{\dagger}_{\uparrow}d^{\dagger}_{\downarrow}+d_{\downarrow}d_{\uparrow}\bigr{\rangle}|=1-Q$. It increases significantly at $E_{A}\simeq U/2$, i.e., near the quarter-filling point $Q=0.5$ ($\delta=\pi/4$) of Bogoliubov particles, and it saturates to the upper-bound value $1.0$ as $E_{A}$ increases further towards the empty-orbital regime.

The Kondo behaviors of Bogoliubov particles are clearly seen for the renormalized Andreev level $\widetilde{E}_{A}$ and the wave-function renormalization factor $Z=\widetilde{\Gamma}_{N}/\Gamma_{N}$, plotted in Figs. 3(c) and 3(d), respectively. The renormalized level is almost locked at the Fermi level $\widetilde{E}_{A}\simeq 0.0$, for large interactions $U/(\pi\Gamma_{N})\gtrsim 2.0$, over a wide Kondo-dominated region $0\leq E_{A}\lesssim U/2$, taking place the inside of the semicircle in Fig. 2. Correspondingly, the renormalization factor $Z$ is significantly suppressed in this region, and it indicates the fact that the Kondo energy scale $T^{*}$,

$\displaystyle T^{*}\,\equiv\,\frac{Z}{4\rho_{d}}\,,\qquad\qquad\rho_{d}\,=\,\frac{\sin^{2}\delta}{\pi\Gamma_{N}}\,,$

(40)

becomes much smaller than the bare tunneling energy scale $\Gamma_{N}$.

In contrast, at $E_{A}\gtrsim U/2$, i.e., in the valence-fluctuation or empty-orbital regime for Bogoliubov particles, the effects of electron correlations become less important: the renormalization factor approaches $Z\simeq 1.0$ and the renormalized level $\widetilde{E}_{A,\sigma}$ approaches the Hartree-Fock (HF) energy shift:

	$\displaystyle E_{A,\sigma}^{\mathrm{HF}}\,\equiv\,E_{A}-\sigma b\,+\,U\left(Q_{\overline{\sigma}}-\frac{1}{2}\right)$		(41)
	$\displaystyle\ \ \xrightarrow{\,E_{A}\gg U/2,\,b=0\,}\,E_{A}\,-\,\frac{U}{2}\,,$

since $\,Q_{\overline{\sigma}}\simeq 0.0$ at $E_{A}\gg U/2$ and $b=0$.

We next discuss the transport properties. Specifically, in this subsection, we consider the case $\Theta=\pi/2$, where $\xi_{d}=0$ and the occupation number of impurity electrons is fixed at $\langle n_{d}\rangle=1$, reflecting the electron-hole symmetry of $H_{\mathrm{eff}}$ defined in Eq. (50). In this case, the Andreev level takes the value $E_{A}=\Gamma_{S}$, which is determined solely by the coupling strength between the QD and the SC lead and it breaks the particle-hole symmetry of the Bogoliubov particles even at $\xi_{d}=0$.

The transmission probability $\mathcal{T}_{\mathrm{ET}}$ of the single-electron tunneling process is shown in Fig. 4(a). We see that the plateau of the unitary limit $\mathcal{T}_{\mathrm{ET}}\simeq 1.0$ evolves at $0\leq E_{A}\lesssim U/2$, for large $U$. Since $\mathcal{T}_{\mathrm{ET}}=\mathcal{T}_{\mathrm{BG}}-\mathcal{T}_{\mathrm{CP}}$ due to the optical theorem mentioned above, it is the Bogoliubov-particle part $\mathcal{T}_{\mathrm{BG}}=\sin^{2}\delta$ that shows the genuine Kondo ridge, as demonstrated in Fig. 4(b). The single-particle contribution $\mathcal{T}_{\mathrm{ET}}$ decreases outside of the Kondo regime $E_{A}\gtrsim U/2$, at which the occupation number $Q$ of Bogoliubov particles rapidly decreases and the SC pair correlation increases, as demonstrated in Figs. 3(a) and 3(b).

The Cooper-pair contribution $\mathcal{T}_{\mathrm{CP}}$ is also plotted in Fig. 4(b), choosing the Bogoliubov angle to be $\Theta=\pi/2$ and multiplying a factor of $-2$ which emerges for the nonlocal conductance $g_{\mathrm{RL}}\propto\mathcal{T}_{\mathrm{BG}}-2\mathcal{T}_{\mathrm{CP}}$: the negative sign represents the fact that the crossed Andreev reflection induces the counterflow, flowing from the right lead towards the QD. In this case, Eq. (36) can be rewritten further into a similar form to the current noise of normal electrons: $\mathcal{T}_{\mathrm{CP}}=\sin^{2}\delta\,(1-\sin^{2}\delta)$. Blanter and Büttiker (2000); Oguri et al. (2022) Thus, the contribution of $\mathcal{T}_{\mathrm{CP}}$ to the nonlocal conductance is maximized in the case at which the phase shift becomes $\delta=\pi/4$ and it reaches the value $-2\mathcal{T}_{\mathrm{CP}}=-\frac{1}{2}$. The corresponding dip emerges in Fig. 4(b) at the crossover region $E_{A}\simeq U/2$, the width of which becomes of the order of $\Gamma_{N}$.

Figure 4(c) shows the nonlocal conductance, which takes the form $g_{\mathrm{RL}}/g_{0}=-2\sin^{2}\delta\cos 2\delta$ at $\Theta=\pi/2$, as mentioned. It decreases from the unitary-limit value $g_{\mathrm{RL}}/g_{0}=1$ as $E_{A}$ deviates from $E_{A}=0$, and vanishes $g_{\mathrm{RL}}=0$ at $E_{A}\simeq U/2$ where the phase shift reaches $\delta=\pi/4$. The nonlocal conductance becomes negative $g_{\mathrm{RL}}<0$ at $E_{A}\gtrsim U/2$ as the CAR contributions dominate in this region. In particular, it has a dip of the depth $g_{\mathrm{RL}}/g_{0}=-1/4$ at the point where the phase shift takes the value $\delta=\pi/6$.

Similarly, the CAR efficiency takes a simplified form $\eta_{\text{CAR}}=\cos^{2}\delta$ at $\Theta=\pi/2$, and the NRG results are plotted in Fig. 4(d). The efficiency $\eta_{\text{CAR}}$ increases with $E_{A}$, and reaches $\eta_{\text{CAR}}=0.5$ at $\delta=\pi/4$ where $-2\mathcal{T}_{\mathrm{CP}}$ has the dip of the depth $-1/2$ seen in Fig. 4(b). The transient region of $\eta_{\text{CAR}}$ varying from 0 to 1 is estimated to be of the order of $\Gamma_{N}$. Furthermore, at $E_{A}\gg U/2$, the efficiency approaches the saturation value $\eta_{\text{CAR}}=1.0$ although the conductance $g_{\mathrm{RL}}$ itself becomes very small.

So far, we have discussed the transport properties at $\Theta=\pi/2$, along the vertical $\Gamma_{S}$ axis in the $\xi_{d}$ vs $\Gamma_{S}$ plane. As $\xi_{d}$ varies from the electron-hole symmetric point $\xi_{d}=0$, the Bogoliubov angle $\Theta$ deviates from $\pi/2$. Here we discuss how the ground-state and transport properties vary along the angular direction over the range $0\leq\Theta\leq\pi$.

Figures 5 and 6 show the NRG results of the renormalized parameters and the transport coefficients as functions of $\xi_{d}$ and $\Gamma_{S}$ for a relatively large Coulomb interaction $U/(\pi\Gamma_{N})=5.0$. In these three-dimensional plots, mesh lines are drawn along the polar coordinates $(E_{A},\Theta)$. Note that the superconducting coherence factors, $\cos\Theta$ and $\sin\Theta$, vary in the angular direction: Cooper pairs are strongly entangled at $\pi/2$ and the SC proximity effect becomes weak as $\Theta$ deviates towards $0$ or $\pi$. In contrast, along the radial direction, the crossover between the Kondo regime and valence fluctuation regime of the Bogoliubov particles occurs near the semicircle of radius $E_{A}=U/2$, as mentioned.