*2.2. Calculation of Currents*

We consider a steady-state current, with the net transport current through the 3QD system, *Itr* = *I*12 + *I*13, expressed as a sum of the bond currents through the upper and the lower branches

$$I\_{i\bar{j}} = \frac{e}{\imath\hbar} (\overline{t}\_{i\bar{j}} \langle c\_i^\dagger c\_{\bar{j}} \rangle - \overline{t}\_{\bar{j}i} \langle c\_{\bar{j}}^\dagger c\_{\bar{i}} \rangle). \tag{5}$$

We use the non-equilibrium Green function technique (NEGF), which is described in many textbooks (e.g., see [31]). To determine the currents, one calculates the lesser Green functions, *G*<*ji* ≡ *<sup>ı</sup>c*†*i cj*, by means of the equation of motion method (EOM). The coupling with the electrodes is manifested by the lesser Green functions *g*<sup>&</sup>lt;*α* = <sup>2</sup>*π*(*g<sup>r</sup>α* − *g<sup>a</sup>α*)*f<sup>α</sup>*, where *g<sup>r</sup>*,*<sup>a</sup> α* denotes the retarded (r) and advanced (a) Green functions in the *α* electrode, and *fα* = 1/(exp[(*<sup>E</sup>* − *μα*)/*kBTα*] + 1) is the Fermi distribution function for an electron with energy *E*, with respect to a chemical potential *μα* and at temperature *T<sup>α</sup>*. For any Green function *G*<*ji* , we separate contributions from the left and the right electrodes (i.e., we extract the coefficients in front of *g*<*L* and *g*<*R* ) and, after some algebra, the bond current can be expressed as

$$I\_{ij} = -\frac{e}{2\pi\hbar} \int\_{-\infty}^{\infty} dE \,\left[\mathcal{G}\_{ij}^{L}(E)f\_L - \mathcal{G}\_{ij}^{R}(E)f\_{\mathbb{R}}\right],\tag{6}$$

where the dimensionless conductances for the upper and the lower branches are

$$\mathcal{G}\_{12}^{L} = 2\Gamma\_L t\_{12} \odot [d\_{23,31} d\_{23,23}^\*] / A\_\prime \tag{7}$$

$$\mathcal{G}\_{12}^{R} = 2\Gamma\_{R} t\_{12} \odot [d\_{23,31}^{\*} d\_{31,31}^{\*}] / A\_{\prime} \tag{8}$$

$$\mathcal{G}\_{13}^L = 2\Gamma\_L t\_{31} \odot [d\_{12,23} d\_{23,23}^\*] / A\_\prime \text{ and} \tag{9}$$

$$\mathcal{G}\_{13}^{K} = 2\Gamma\_R t\_{31} \odot [d\_{12,31} d\_{23,31}] / A. \tag{10}$$

Here, we denote the coefficients: *d*12,23 = *<sup>e</sup>*<sup>−</sup>*<sup>ı</sup>φt*12*t*23 − *<sup>t</sup>*31*wr*2, *d*12,31 = *t*12*t*31 − *<sup>e</sup>*<sup>−</sup>*<sup>ı</sup>φt*23*wr*1, *d*23,31 = *<sup>e</sup><sup>ı</sup>φt*23*t*31 − *t*12*w*3, *d*23,23 = *t*223 − *<sup>w</sup>r*2*w*3, *d*31,31 = *t*231 − *<sup>w</sup>r*1*w*3, and the denominator

$$A = \left| w\_1^r t\_{23}^2 + w\_2^r t\_{31}^2 + w\_3 t\_{12}^2 - w\_1^r w\_2^r w\_3 - 2t\_{12} t\_{23} t\_{31} \cos \phi \right|^2,\tag{11}$$

where *wr*,*<sup>a</sup>* 1 = *E* − *ε*1 − *γ<sup>r</sup>*,*<sup>a</sup> L* , *wr*,*<sup>a</sup>* 2 = *E* − *ε*2 − *γ<sup>r</sup>*,*<sup>a</sup> R* , *w*3 = *E* − *ε*3, Γ*α* = <sup>2</sup>[*γ<sup>a</sup>α*]*t*2*α*, and *γ<sup>r</sup>*,*<sup>a</sup> α* = *g<sup>r</sup>*,*<sup>a</sup> α t*2*α*.

Note that Equation (6) includes the transport current due to the bias voltage applied to the electrodes, as well as the persistent current induced by the magnetic flux [a term proportional to sin *φ*], which can be written as *I*12 = *Itr*12 − *Iφ* and *I*13 = *Itr*13 + *<sup>I</sup>φ*, respectively. These coefficients are coupled with those in (7)–(10):

$$\mathcal{G}\_{12}^{L} = \mathcal{G}\_{12} - \mathcal{G}\_{\Phi}^{L}, \ \mathcal{G}\_{12}^{R} = \mathcal{G}\_{12} + \mathcal{G}\_{\Phi}^{R},\tag{12}$$

$$\mathcal{G}^{\rm L}\_{13} = \mathcal{G}\_{13} + \mathcal{G}^{\rm L}\_{\phi'} \quad \mathcal{G}^{R}\_{13} = \mathcal{G}\_{13} - \mathcal{G}^{R}\_{\phi}. \tag{13}$$

The first part is

$$I\_{ij}^{tr} = -\frac{\varepsilon}{2\pi\hbar} \int\_{-\infty}^{\infty} dE \, (f\_L - f\_R) \mathcal{G}\_{ij}(E),\tag{14}$$

where the bond conductances are

$$\mathcal{G}\_{12} = \Gamma\_L \Gamma\_R t\_{12} w\_3 [t\_{23} t\_{31} \cos \phi - t\_{12} w\_3] / A,\text{ and} \tag{15}$$

$$\mathcal{G}\_{13} = \Gamma\_L \Gamma\_R t\_{23} [t\_{12} t\_{31} w\_3 \cos \phi - t\_{23} t\_{31}^2] / A. \tag{16}$$

The net transport current is *Itr* = *Itr*12 + *Itr*13and the transmission is given by

$$\mathcal{T} \equiv \mathcal{G}\_{12} + \mathcal{G}\_{13} = \Gamma\_L \Gamma\_R [2t\_{12}t\_{23}t\_{31}w\_3 \cos \phi - t\_{12}^2 w\_3^2 - t\_{23}^2 t\_{31}^2]/A. \tag{17}$$

The persistent current is expressed as

$$I^{\Phi} \equiv -\frac{e}{\pi \hbar} \int\_{-\infty}^{\infty} dE \,\left(\mathcal{G}\_{\Phi}^{L} f\_{L} + \mathcal{G}\_{\Phi}^{R} f\_{R}\right) \, , \tag{18}$$

where

$$\mathcal{G}\_{\phi}^{L} = \Gamma\_{L} t\_{12} t\_{23} t\_{31} \sin \phi \left[ 2t\_{23}^{2} - (w\_{2}^{a} + w\_{2}^{r}) w\_{3} \right] / A\_{\prime} \text{ and} \tag{19}$$

$$\mathcal{G}\_{\phi}^{R} = \Gamma\_{R} t\_{12} t\_{23} t\_{31} \sin \phi \left[ 2t\_{31}^{2} - (w\_{1}^{a} + w\_{1}^{r}) w\_{3} \right] / A. \tag{20}$$

In the next section, we will show that the voltage bias can induce the circular current, where the bond conductances G*ij* are larger than unity or negative.
