Appendix A
1.1 A.1 Equations of motion for Green’s functions of dots
(a) For the single quantum dot
$$\begin{aligned}&\omega \langle \langle c_{\sigma },c_{\sigma }^{\dag }\rangle \rangle ^{r}=1+\varepsilon \langle \langle c_{\sigma },c_{\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad +U\langle \langle n_{-\sigma }c_{\sigma },c_{\sigma }^{\dag }\rangle \rangle ^{r}+V_{k}^{\mathrm{L}}{\langle \langle a}_{k\sigma },c_{\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad +V_{p}^{\mathrm{R}}{\langle \langle b}_{p\sigma },c_{\sigma }^{\dag }\rangle \rangle ^{r} \end{aligned}$$
(A.1)
Further,
$$\begin{aligned} \langle \langle a_{k\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}=V_{k}^{\mathrm{L}*}g_{k}^{r}\langle \langle c_{\sigma },c_{\sigma }^{\dag }\rangle \rangle ^{r} \end{aligned}$$
and
$$\begin{aligned} \langle \langle b_{p\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}=V_{p}^{\mathrm{R}*}g_{k}^{r}\langle \langle c_{\sigma },c_{\sigma }^{\dag }\rangle \rangle ^{r}. \end{aligned}$$
(b) For the double quantum dot in series
$$\begin{aligned}&\left( \omega -\epsilon _{1} \right) \langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}=1+V_{d}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad +U_{1}\langle \langle c_{1\sigma }n_{1-\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}+V_{k}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r} \end{aligned}$$
(A.2)
$$\begin{aligned}&\left( \omega -\epsilon _{2} \right) \langle \langle c_{2\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}=1+V_{d}^{\mathrm{*}}\langle \langle c_{1\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad +U_{2}\langle \langle c_{2\sigma }n_{2-\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r} +V_{p}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r} \end{aligned}$$
(A.3)
$$\begin{aligned}&\left( \omega -\epsilon _{2} \right) \langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}=V_{d}^{\mathrm{*}}\langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad +U_{2}\langle \langle c_{2\sigma }n_{2-\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}+V_{p}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r} \end{aligned}$$
(A.4)
$$\begin{aligned}&\left( \omega -\epsilon _{1} \right) \langle \langle c_{1\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}=V_{d}\langle \langle c_{2\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad +U_{1}\langle \langle c_{1\sigma }n_{1-\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}+V_{k}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}.\nonumber \\ \end{aligned}$$
(A.5)
Here
$$\begin{aligned} \langle \langle a_{k\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}=V_{k1}^{\mathrm{L}*}g_{k}^{r}\langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r} \end{aligned}$$
and
$$\begin{aligned} \langle \langle b_{p\sigma },c_{1\sigma }^{\dag }{\rangle \rangle }^{r}=V_{p2}^{\mathrm{R}*}g_{k}^{r}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}. \end{aligned}$$
(c) For the double quantum dot in parallel
$$\begin{aligned}&\left( \omega -\epsilon _{1} \right) \langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}=1+V_{d}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +U_{1}\langle \langle c_{1\sigma }n_{1-\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}+V_{k1}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r} \nonumber \\&\qquad +V_{p1}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{{r}} \end{aligned}$$
(A.6)
$$\begin{aligned}&\left( \omega -\epsilon _{1}-n_{1}U_{1} \right) \langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}=1+V_{d}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +V_{k1}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{1\sigma }^{\dag }{\rangle \rangle }^{{r}}+V_{p1}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r} \end{aligned}$$
(A.7)
$$\begin{aligned}&\left( \omega -\epsilon _{1}-n_{1}U_{1}-i\frac{\Gamma _{11}}{2} \right) \langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad =1+V_{d}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}+i\frac{\Gamma _{12}}{2}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r} \end{aligned}$$
(A.8)
$$\begin{aligned}&\left( \omega -\epsilon _{2}-n_{2}U_{2}-i\frac{\Gamma _{22}}{2} \right) \langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad =V_{d}\langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}+i\frac{\Gamma _{21}}{2}\langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r} \end{aligned}$$
(A.9)
$$\begin{aligned}&\left( \omega -\epsilon _{2}-n_{2}U_{2}-i\frac{\Gamma _{22}}{2} \right) \langle \langle c_{2\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad =1+V_{d}\langle \langle c_{1\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}+i\frac{\Gamma _{21}}{2}\langle \langle c_{1\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r} \end{aligned}$$
(A.10)
$$\begin{aligned}&\left( \omega -\epsilon _{1}-n_{1}U_{1} \right) \langle \langle c_{1\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad =1+V_{d}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}+V_{k1}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +V_{p1}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r} \end{aligned}$$
(A.11)
$$\begin{aligned}&\left( \omega -\epsilon _{1}-n_{1}U_{1}-i\frac{\Gamma _{11}}{2} \right) \langle \langle c_{1\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad =1+V_{d}\langle \langle c_{2\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}+i\frac{\Gamma _{12}}{2}\langle \langle c_{2\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}.\nonumber \\ \end{aligned}$$
(A.12)
The Green’s functions \(\langle \langle a_{k\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\) and \(\langle \langle b_{p\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\) arise due to the tunnelling of electrons between dots and leads. The EOM for these gives the following Dyson equations:
$$\begin{aligned} \langle \langle a_{k\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}= & {} V_{k1}^{\mathrm{L}*}g_{k}^{\mathrm{r}}\langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\\&+V_{k2}^{\mathrm{L}*}g_{k}^{r}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\\ \langle \langle b_{p\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}= & {} V_{p1}^{\mathrm{R}*}g_{k}^{r}\langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\\&+V_{p2}^{\mathrm{R}*}g_{k}^{r}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}. \end{aligned}$$
(d) For the triple quantum dots in series
$$\begin{aligned}&\left( \omega -\epsilon _{1} \right) \langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}=1+V_{d}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad +U_{1}\langle \langle c_{1\sigma }n_{1-\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}+V_{k}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r} \end{aligned}$$
(A.13)
$$\begin{aligned}&\left( \omega -\epsilon _{2} \right) \langle \langle c_{2\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}=1+V_{d}^{*}\langle \langle c_{1\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad +V_{d}\langle \langle c_{3\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}+U_{2}\langle \langle c_{2\sigma }n_{2-\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\end{aligned}$$
(A.14)
$$\begin{aligned}&\left( \omega -\epsilon _{3} \right) \langle \langle c_{3\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}=1+V_{d}^{*}\langle \langle c_{2\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad +U_{3}\langle \langle c_{3\sigma }n_{3-\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}+V_{p}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r} \end{aligned}$$
(A.15)
$$\begin{aligned}&\left( \omega -\epsilon _{1} \right) \langle \langle c_{1\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}=V_{d}\langle \langle c_{2\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad +U_{1}\langle \langle c_{1\sigma }n_{1-\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}+V_{k}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\end{aligned}$$
(A.16)
$$\begin{aligned}&\left( \omega -\epsilon _{1} \right) \langle \langle c_{1\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}=V_{d}\langle \langle c_{2\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad +U_{1}\langle \langle c_{1\sigma }n_{1-\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}+V_{k}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}\end{aligned}$$
(A.17)
$$\begin{aligned}&\left( \omega -\epsilon _{2} \right) \langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}=V_{d}^{*}\langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad +V_{d}\langle \langle c_{3\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}+U_{2}\langle \langle c_{2\sigma }n_{2-\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\end{aligned}$$
(A.18)
$$\begin{aligned}&\left( \omega -\epsilon _{2} \right) \langle \langle c_{2\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}=V_{\mathrm{d}}^{\mathrm{*}}\langle \langle c_{1\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad +V_{d}\langle \langle c_{3\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}+U_{2}\langle \langle c_{2\sigma }n_{2-\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r} \end{aligned}$$
(A.19)
$$\begin{aligned}&\left( \omega -\epsilon _{3} \right) \langle \langle c_{3\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}=V_{d}^{*}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad +U_{3}\langle \langle c_{3\sigma }n_{3-\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}+V_{p}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\end{aligned}$$
(A.20)
$$\begin{aligned}&\left( \omega -\epsilon _{3} \right) \langle \langle c_{3\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}=V_{d}^{*}\langle \langle c_{2\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad +U_{3}\langle \langle c_{3\sigma }n_{3-\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}+V_{p}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}. \end{aligned}$$
(A.21)
The Dyson equations for lead-dot GFs in series combination are
$$\begin{aligned} \langle \langle a_{k\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}=V_{k1}^{\mathrm{L}*}g_{k}^{r}\langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r} \end{aligned}$$
and
$$\begin{aligned} \langle \langle b_{p\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}=V_{p3}^{\mathrm{R}*}g_{k}^{\mathrm{r}}\langle \langle c_{3\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}. \end{aligned}$$
(e) For the triple quantum dots in parallel
$$\begin{aligned}&\left( \omega -\epsilon _{1} \right) \langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}=1+V_{d}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +U_{1}\langle \langle c_{1\sigma }n_{1-\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}+V_{k1}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +V_{p1}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\end{aligned}$$
(A.22)
$$\begin{aligned}&\left( \omega -\epsilon _{2} \right) \langle \langle c_{2\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}=1+V_{d}^{*}\langle \langle c_{1\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +V_{d}\langle \langle c_{3\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}+V_{k2}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r} \nonumber \\&\qquad +U_{2}\langle \langle c_{2\sigma }n_{2-\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{{r}}+ V_{p2}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\end{aligned}$$
(A.23)
$$\begin{aligned}&\left( \omega -\epsilon _{3} \right) \langle \langle c_{3\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}=1+V_{d}^{*}\langle \langle c_{2\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +U_{3}\langle \langle c_{3\sigma }n_{3-\sigma },c_{3\sigma }^{\dag }{\rangle \rangle }^{{r}}+V_{k3}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +V_{p3}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}\end{aligned}$$
(A.24)
$$\begin{aligned}&\left( \omega -\epsilon _{1} \right) \langle \langle c_{1\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}=V_{\mathrm{d}}\langle \langle c_{2\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +U_{1}\langle \langle c_{1\sigma }n_{1-\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}+V_{k1}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +V_{p1}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\end{aligned}$$
(A.25)
$$\begin{aligned}&\left( \omega -\epsilon _{2} \right) \langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}=V_{d}^{*}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +V_{d}\langle \langle c_{3\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}+U_{2}\langle \langle c_{2\sigma }n_{2-\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad + V_{p2}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{{r}}+ V_{k2}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\end{aligned}$$
(A.26)
$$\begin{aligned}&\left( \omega -\epsilon _{3} \right) \langle \langle c_{3\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}=V_{d}^{*}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +U_{3}\langle \langle c_{3\sigma }n_{3-\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}+V_{k3}^{\mathrm{L}} \langle \langle a_{k\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +V_{p3}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\end{aligned}$$
(A.27)
$$\begin{aligned}&\left( \omega -\epsilon _{1} \right) \langle \langle c_{1\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}=V_{d}\langle \langle c_{2\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +U_{1}\langle \langle c_{1\sigma }n_{1-\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}+V_{k1}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +V_{p1}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}\end{aligned}$$
(A.28)
$$\begin{aligned}&\left( \omega -\epsilon _{2} \right) \langle \langle c_{2\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}=V_{d}^{*}\langle \langle c_{1\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +V_{d}\langle \langle c_{3\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}+U_{2}\langle \langle c_{2\sigma }n_{2-\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad + V_{p2}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}+ V_{k2}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}\end{aligned}$$
(A.29)
$$\begin{aligned}&\left( \omega -\epsilon _{3} \right) \langle \langle c_{3\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}=V_{d}^{*}\langle \langle c_{2\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +U_{3}\langle \langle c_{3\sigma }n_{3-\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}+V_{k3}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad +V_{p3}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{2\sigma }^{\dag }\rangle \rangle ^{r}\end{aligned}$$
(A.30)
$$\begin{aligned}&\left( \omega -\epsilon _{1}-n_{1}U_{1}-i\frac{\Gamma _{11}}{2} \right) \langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad =1+V_{d}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}+i\frac{\Gamma _{\mathrm{12}}}{2}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad + i\frac{\Gamma _{\mathrm{13}}}{2}\langle \langle c_{3\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\end{aligned}$$
(A.31)
$$\begin{aligned}&\left( \omega -\epsilon _{2}-n_{2}U_{2}-i\frac{\Gamma _{22}}{2} \right) \langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\quad =V_{d}\langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}+i\frac{\Gamma _{21}}{2}\langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\end{aligned}$$
(A.32)
$$\begin{aligned}&\left( \omega -\epsilon _{3}-n_{3}U_{3}-i\frac{\Gamma _{33}}{2} \right) \langle \langle c_{3\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}\nonumber \\&\qquad =1+V_{d}\langle \langle c_{2\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}+i\frac{\Gamma _{32}}{2}\langle \langle c_{2\sigma },c_{3\sigma }^{\dag }\rangle \rangle ^{r}.\nonumber \\ \end{aligned}$$
(A.33)
The EOM for lead-dot coupling GFs are given by following Dyson equations:
$$\begin{aligned} \langle \langle a_{k\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}= & {} V_{k1}^{\mathrm{L}*}g_{k}^{r}\langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\\&+V_{k2}^{\mathrm{L}*}g_{k}^{{r}}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\\&+V_{k3}^{\mathrm{L}*}g_{k}^{r}\langle \langle c_{3\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\\ \langle \langle b_{p\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}= & {} V_{p1}^{\mathrm{R}*}g_{{k}}^{r}{\langle \langle }c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\\&+V_{p2}^{\mathrm{R}*}g_{k}^{r}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}\\&+V_{p3}^{\mathrm{R}*}g_{k}^{{r}}\langle \langle c_{3\sigma },c_{1\sigma }^{\dag }\rangle \rangle ^{r}. \end{aligned}$$
The GFs of QDs in various configurations can be obtained by using the simplest mean-field approximation, such that GFs of the type \(\langle \langle c_{i\sigma }n_{i-\sigma },c_{j\sigma }^{\dag }{\rangle \rangle }^{{r}}\approx n_{i-\sigma }\langle \langle c_{i\sigma },c_{j\sigma }^{\dag }{\rangle \rangle }^{{r}}\) and hierarchy of GFs in the above EOMs get closed to the final forms of the respective GFs.
Appendix B
1.1 B.1 Current
To discuss the method for deriving relation for the total current, we take the case of parallel double quantum dot system. The expression for the current due to the left lead is
$$\begin{aligned}&I_{\mathrm{L}}=\frac{e}{h}\mathop {\sum }\limits _{ki\sigma } \int {\left( V_{ki}^{\mathrm{L}}\langle \langle a_{k\sigma },c_{i\sigma }^{\dag }\rangle \rangle _{\omega }^{<}-V_{ki}^{\mathrm{L}*}\langle \langle c_{k\sigma },a_{i\sigma }^{\dag }\rangle \rangle _{\omega }^{<} \right) \mathrm {d}\omega } \nonumber \\&\quad \quad (i =1, 2). \end{aligned}$$
(B.1)
Similarly, the current due to the right lead can be written as
$$\begin{aligned}&I_{\mathrm{R}}=\frac{e}{h}\mathop {\sum }\limits _{pi\sigma } \int {\left( V_{pi}^{\mathrm{R}}\langle \langle b_{p\sigma },c_{i\sigma }^{\dag }\rangle \rangle _{\omega }^{<}-V_{pi}^{\mathrm{R}*}\langle \langle c_{k\sigma },b_{i\sigma }^{\dag }\rangle \rangle _{\omega }^{<} \right) \mathrm {d}\omega } \nonumber \\&\quad \quad (i=1, 2). \end{aligned}$$
(B.2)
The Dyson equations for the GFs appearing in these expressions are
$$\begin{aligned}&\langle \langle a_{k\sigma },c_{1\sigma }^{\dag }\rangle \rangle _{\omega }^{<}\nonumber \\&\quad =V_{k1}^{\mathrm{L}*}g_{k}^{{r}}\langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle _{\omega }^{<}+V_{k1}^{\mathrm{L}*}g_{k}^{<}\langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle _{\omega }^{a}\nonumber \\&\qquad + {V}_{k2}^{\mathrm{L}*}g_{k}^{{r}}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle _{\omega }^{<}+V_{k2}^{\mathrm{L}*}g_{k}^{<}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle _{\omega }^{a}\nonumber \\&\langle \langle a_{k\sigma },c_{2\sigma }^{\dag }\rangle \rangle _{\omega }^{<}\nonumber \\&\quad =V_{k2}^{\mathrm{L}*}g_{k}^{{r}}\langle \langle c_{2\sigma },c_{2\sigma }^{\dag }\rangle \rangle _{\omega }^{<}+V_{k2}^{\mathrm{L}*}g_{k}^{<}\langle \langle c_{2\sigma },c_{2\sigma }^{\dag }\rangle \rangle _{\omega }^{a}\nonumber \\&\qquad + {{V}}_{{k1}}^{\mathrm{L}*}g_{k}^{{r}}\langle \langle c_{1\sigma },c_{2\sigma }^{\dag }\rangle \rangle _{\omega }^{<}+V_{k1}^{\mathrm{L}*}g_{k}^{<}\langle \langle c_{1\sigma },c_{2\sigma }^{\dag }\rangle \rangle _{\omega }^{a}\nonumber \\&\langle \langle c_{1\sigma },a_{k\sigma }^{\dag }\rangle \rangle _{\omega }^{<}\nonumber \\&\quad =V_{k1}^{\mathrm{L}}g_{k}^{<}\langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle _{\omega }^{r}+V_{k1}^{\mathrm{L}}g_{k}^{a}\langle \langle c_{1\sigma },c_{1\sigma }^{\dag }\rangle \rangle _{\omega }^{<}\nonumber \\&\qquad + {V}_{k2}^{\mathrm{L}}g_{k}^{<}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle _{\omega }^{r}+V_{k2}^{\mathrm{L}}g_{k}^{a}\langle \langle c_{2\sigma },c_{1\sigma }^{\dag }\rangle \rangle _{\omega }^{<}\nonumber \\&\langle \langle c_{2\sigma },a_{k\sigma }^{\dag }\rangle \rangle _{\omega }^{<}\nonumber \\&\quad =V_{k2}^{\mathrm{L}}g_{k}^{<}\langle \langle c_{2\sigma },c_{2\sigma }^{\dag }\rangle \rangle _{\omega }^{r}+V_{k2}^{\mathrm{L}}g_{k}^{a}\langle \langle c_{2\sigma },c_{2\sigma }^{\dag }\rangle \rangle _{\omega }^{<}\nonumber \\&\qquad + {V}_{k1}^{\mathrm{L}}g_{k}^{<}\langle \langle c_{1\sigma },c_{2\sigma }^{\dag }{\rangle \rangle }_{{\omega }}^{r}+V_{k1}^{\mathrm{L}}g_{k}^{a}\langle \langle c_{1\sigma },c_{2\sigma }^{\dag }\rangle \rangle _{\omega }^{<} \nonumber \\&I_{\mathrm{L}}=\frac{e}{h}\mathop {\sum }\limits _\sigma \int {i\mathrm {Tr}\left[ \Gamma ^{\mathrm{L}}G^{<}+f_{\mathrm{L}}\Gamma ^{\mathrm{L}}(G^{r}-G^{a}) \right] \mathrm {d}\omega } \end{aligned}$$
(B.3)
$$\begin{aligned}&I_{\mathrm{R}}=\frac{e}{h}\mathop {\sum }\limits _\sigma \int {i\mathrm {Tr}\left[ \Gamma ^{\mathrm{R}}G^{<}+f_{\mathrm{R}}\Gamma ^{\mathrm{R}}(G^{r}-G^{a}) \right] \mathrm{d}\omega .}\nonumber \\ \end{aligned}$$
(B.4)
Further,
$$\begin{aligned} I=I_{\mathrm{L}}=-I_{\mathrm{R}}=\frac{I_{\mathrm{L}}-\mathrm {I}_{\mathrm{R}}}{2}. \end{aligned}$$
The total current in the couple quantum dots is
$$\begin{aligned} I= & {} \frac{ie}{2h}\mathop {\sum }\limits _\sigma \int \mathrm {Tr}\left[ \left( \Gamma ^{\mathrm{L}}-\Gamma ^{\mathrm{R}} \right) G^{<}\right. \nonumber \\&\left. +({\Gamma ^{\mathrm{L}}f}_{\mathrm{L}}{-\Gamma ^{\mathrm{R}}f}_{\mathrm{R}})(G^{r}-G^{a}) \right] \mathrm {d}\omega . \end{aligned}$$
(B.5)
For the proportionate case
$$\begin{aligned} \Gamma ^{\mathrm{L}}=\lambda \Gamma ^{\mathrm{R}}\quad \hbox { and }\quad x=\frac{1}{1+\lambda }. \end{aligned}$$
The current can be written as \(I=xI_{\mathrm{L}}-(1-x)I_{\mathrm{R}}\). Hence,
$$\begin{aligned} I= & {} \frac{ie}{h}\mathop {\sum }\limits _\sigma \int (f_{\mathrm{L}}{-f}_{\mathrm{R}})\nonumber \\&\times \mathrm {Tr}\left[ \left( \frac{\Gamma ^{\mathrm{L}}\Gamma ^{\mathrm{R}}}{\Gamma ^{\mathrm{L}}-\Gamma ^{\mathrm{R}}} \right) (G^{r}-G^{a}) \right] \mathrm {d}\omega . \end{aligned}$$
(B.6)
Also, by making use of the identity \(G^{r}-G^{a}=-i\left( \Gamma ^{\mathrm{L}}-\Gamma ^{\mathrm{R}} \right) G^{r}G^{a}\), the expression for the total current leads to the following well-known Landauer–Buttiker current formula:
$$\begin{aligned} I= & {} \frac{e}{h}\mathop {\sum }\limits _\sigma \int {\left( f_{\mathrm{L}}-f_{\mathrm{R}} \right) \mathrm {Tr}\left( G^{a}\Gamma ^{\mathrm{R}}G^{{r}}\Gamma ^{\mathrm{L}} \right) \mathrm {d}\omega }\nonumber \\= & {} \frac{e}{h}\mathop {\sum }\limits _\sigma \int {\left( f_{\mathrm{L}}-f_{\mathrm{R}} \right) T\left( \omega \right) \mathrm {d}\omega }. \end{aligned}$$
(B.7)
