**6. Summary**

We considered the influence of quantum interference on electron transport and current correlations in a ring of three quantum dots threaded by a magnetic flux. We assumed non-interacting electrons and calculated the bond conductances, the local currents, and the current correlation functions—in particular, the shot noise—by means of the non-equilibrium Keldysh Green function technique, taking into account multiple reflections of the electron wave inside the ring. As we considered elastic scatterings, for which Kirchhoff's current law is fulfilled, the transmission T = G12 + G13 is a sum of the local bond conductances and the shot noise for the transport current is a composition of the local current correlation functions, <sup>S</sup>*shtr*,*tr*= <sup>S</sup>*sh*12,12+ <sup>S</sup>*sh*13,13+ <sup>2</sup>S*sh*12,13= T (1 − T ), which gives the Lesovik formula.

In the system, having triangular symmetry, the eigenstates *E*0 = −2 and *E*± = 1 (with the wavevector *k* = 0 and *k* = <sup>±</sup>2*π*/3) play a different role in the transport, which is seen in the bond conductances and the shot noise components. An electron wave injected with energy close to *E*0 is perfectly split into both branches of the ring and the current cross-correlation function <sup>S</sup>*sh*12,13 is positive. At the resonance *E*0, the transmission T = 1 and all correlation functions <sup>S</sup>*sh*12,12 = <sup>S</sup>*sh*13,13 = <sup>S</sup>*sh*12,13 = 0, which means that the bond currents are uncorrelated. The magnetic flux changes quantum interference conditions and correlates the bond currents; the cross-correlation <sup>S</sup>*sh*12,13 becomes negative at the resonance and fully compensates the positive auto-correlation components <sup>S</sup>*sh*12,12 and <sup>S</sup>*sh*13,13 (with <sup>S</sup>*shtr*,*tr* = 0).

Quantum interference plays a crucial role in transport through the degenerate states at *E*± = 1, where one can observe Fano resonance with destructive interference. In this region, the circular current *Idr* can be driven by the bias voltage. The bond conductances have an opposite sign, their maximal value is inversely proportional to the coupling, Γ*<sup>α</sup>*, with the electrodes, and they can be larger than unity. The direction of *Idr* depends on the bias voltage and the position of the Fermi energy *EF*, with respect to the degenerate state *E*±. The auto-correlation functions <sup>S</sup>*sh*12,12, <sup>S</sup>*sh*13,13 are large (inversely proportional to <sup>Γ</sup>2*α*) close to the resonance. The cross-correlator <sup>S</sup>*sh*12,13 is negative in the presence of the driven circular current. Our calculations show that a small magnetic flux, *φ* = 2*π*/16, can destroy the Fano resonance, and two resonance peaks (with T = 1) appear. The driven component, *<sup>I</sup>dr*, is reduced with an increase of *φ*, and it disappears at *φ* = 2*π*/4. However, quantum interference still plays a role; the bond currents are strongly correlated (with large <sup>S</sup>*sh*12,12 and <sup>S</sup>*sh*13,13 and negative <sup>S</sup>*sh*12,13). For a large coupling, the driven part *Idr* can be large and can profoundly modify the total circular current *Ic* = *Idr* + *Iφ*.

We also performed calculations of the bond currents and their correlations for rings with a various number of sites; in particular, for the benzene ring in para-, metha-, and ortho-connection with the electrodes. The results are qualitatively similar to those presented above for the 3QD ring: Quantum interference of the travelling waves with the eigenstates of opposite chirality leads to the driven circular currents, accompanied by large current fluctuations with a negative cross-correlation component. To observe this effect, the two conducting branches should be asymmetric; in particular, in the benzene ring, the driven circular current appears for the metha- and ortho-connections, but is absent in the para-connection, where both conducting branches are equivalent (see also [7]).

An open problem is including interactions between electrons into the calculations of the coherent transport and shot noise. Coulomb interactions can be taken into account in the sequential regime [52], or by using the real-time diagrammatic technique [53–55]; however, in practice, one includes only firstand second-order diagrams with respect to the tunnel coupling and the role of QI is diminished. In principle, one can treat QI on an equal footing with electron interactions in the framework of quantum field theory [56], as was done for the Anderson single impurity model, by means of full counting statistics (FCS), where the average current and all its moments were calculated [57]. However, this is a formidable task, even for the simple 3QD model.

**Author Contributions:** Both the authors have a similar contribution to the paper in its concept, research, and manuscript preparation.

**Funding:** The research was financed by National Science Centre, Poland—project number 2016/21/B/ST3/02160. **Conflicts of Interest:** The authors declare no conflict of interest.

### **Appendix A. Coupling to Atomic Chain Electrodes: Analytical Results**

The results for the conductances and shot noise may be simplified when we take all hopping integrals equal to *t*, the same position of the site levels *ε* = 0, and the symmetric coupling *tL* = *tR* = *t*, with the electrodes as a semi-infinite atomic chain. In this case, the Green functions in the electrodes are *gr* = *eık*/*<sup>t</sup>* and *ga* = *e*<sup>−</sup>*ık*/*<sup>t</sup>* and the electron spectrum is *Ek* = 2*t* cos *k*. From Equations (7)–(10), one can calculate the dimensionless bond conductances as

$$\mathcal{G}\_{12}^{L} = 2\sin k \left[ \sin k + \sin 3k - \sin(2k + \phi) \right] / A \,, \tag{A1}$$

$$\mathcal{G}\_{12}^{\mathbb{R}} = 2\sin k \left[ \sin k + \sin 3k - \sin (2k - \phi) \right] / A \,, \tag{A2}$$

$$\mathcal{G}\_{13}^L = 2\sin k \left[ \sin k - \sin(2k - \phi) \right] / A \text{ , and} \tag{A3}$$

$$\mathcal{G}\_{13}^{R} = 2\sin k \left[ \sin k - \sin(2k + \phi) \right] / A \,, \tag{A4}$$

where the denominator

$$A = 4 + \cos 2\phi - 2\cos\phi (3\cos k - \cos 3k) - \cos 4k. \tag{A5}$$

It is seen an asymmetry with respect to the direction of the magnetic flux (to *φ*) for the conductances G*Lij*and G*Rij*from the left and the right electrode. The transmission, T , is expressed as

$$\mathcal{T} \equiv \mathcal{G}\_{12}^{L} + \mathcal{G}\_{13}^{L} = \mathcal{G}\_{12}^{R} + \mathcal{G}\_{13}^{R} = \mathcal{G}\_{12} + \mathcal{G}\_{13} = 2 \sin^{2} k [1 - 4 \cos k (\cos \phi - \cos k)] / A,\tag{A6}$$

where the driven part of the bond conductances are calculated using Equations (15) and (16)

$$\mathcal{G}\_{12} = 4 \sin^2 k \cos k (2 \cos k - \cos \phi) / A \,, \tag{A7}$$

$$\mathcal{G}\_{13} = 2\sin^2 k (1 - 2\cos\phi\cos k) / A \,, \text{ and} \tag{A8}$$

$$\mathcal{G}\_{\Phi}^{L} = \mathcal{G}\_{\Phi}^{R} = 2 \sin \phi \sin k \cos 2k / A \,\tag{A9}$$

and, from Equations (19) and (20), the part induced by the flux is

$$\mathcal{G}\_{\phi}^{L} = \mathcal{G}\_{\phi}^{R} = 2 \sin \phi \sin k \cos 2k \,/\,\text{A} \,\text{.}\tag{A10}$$

It can be seen that the conductance G12 becomes negative at *k* = *π*/2 (i.e., when the circular current becomes driven).

The shot noise for the bond currents is expressed as

$$S\_{12,12}^{\text{sl}} = 4\sin^2 k \left| \epsilon^{\mu \dot{\rho}} (\cos \phi - 2\cos k) + \cos 2k \right|^2 / A^2 \tag{A11}$$

$$S\_{13,13}^{\text{sl}} = 4 \sin^2 k |\cos k - e^{\psi}|^2 / A^2 \,\text{.}\tag{A12}$$

$$\mathcal{S}\_{12,13}^{sh} = -4\sin^2 k \left[ 2\cos\phi \cos k(\cos\phi - \cos k)^2 + \sin^2\phi \right] / A^2 \text{, and} \tag{A13}$$

$$\mathcal{S}\_{tr,tr}^{\rm dh} = \mathcal{T}(1 - \mathcal{T}) = 4 \sin^2 k (\cos \phi - \cos k)^2 [1 - 4 \cos k (\cos \phi - \cos k)] / A^2. \tag{A14}$$

Notice that the cross-correlation <sup>S</sup>*sh*12,13 can be positive or negative in the laminar or the vortex regime, respectively.
