Magnetotransport in Aharonov-Bohm interferometers: Exact numerical simulations

The linear conductance of a two-terminal Aharonov-Bohm interferometer is an even function of the applied magnetic flux, as dictated by the Onsager-Casimir symmetry. Away from linear response this symmetry may be broken when many-body interactions are in effect. Using a numerically exact simulation tool, we study the dynamics and the steady-state behavior of the out-of-equilibrium double-dot Aharonov-Bohm interferometer, while considering different types of interactions: Model I includes a closed interferometer with an interdot electron-electron repulsion energy. In model II the interferometer is interacting with a dissipative environment, possibly driven away from equilibrium. In both cases we show that depending on the (horizontal, vertical) mirror symmetries of the setup, nonlinear transport coefficients obey certain magnetosymmetries. We compare numerically exact simulations to phenomenological approaches and special limits: The behavior of model I is compared to self-consistent mean-field calculations and master equation results in the Coulomb blockade regime. Model II, allowing heat dissipation to a thermal bath, is mimicked by an Aharonov-Bohm junction with a voltage probe. In both cases we find that phenomenological treatments capture the relevant transport symmetries, yet significant deviations in magnitude may show up.

Figure 1

Scheme of model I, a two-terminal double-dot Aharonov-Bohm interferometer with spinless electrons in two quantum dots, 1 and 2, with an interdot repulsion of strength $U$.

Figure 2

Scheme of model II, a two-terminal double-dot Aharonov-Bohm interferometer coupled to a fermionic environment. This environment consists of a quantum dot (labeled $p)$ itself hybridized with either (a) an equilibrium sea of noninteracting electrons, or (b) two metals $(\pm )$, possibly biased away from equilibrium. In both cases dot 1 of the interferometer is coupled capacitively (strength $U)$ to dot $p$ in the fermionic environment.

Figure 3

Schemes of the different setups considered in this work. (a) Model I with a horizontal (H) mirror symmetry. The bold (light) arrows represent strong (weak) hybridization energies of the quantum dots to the metals. Simulations are included in Fig. 4. (b) Model I with a vertical (V) mirror symmetry. The nondegenerate dots are represented by different colors. Figure 5 displays the transients; Fig. 8 shows steady-state values. (c) Model I missing the H and V mirror symmetries; see data in Figs. 7 and 8. (d) Model II with a V mirror symmetry. Results are displayed in Figs. 11 and 13. (e) Model II missing both the H and V mirror symmetries. Steady-state data are included in Fig. 13.

Figure 4

Model I with a horizontal mirror symmetry, showing phase rigidity. INFPI simulations using $\varphi =\pm \pi /2,{\gamma }_{L,1}={\gamma }_{L,2}=0.2,{\gamma }_{R,1}={\gamma }_{R,2}=0.05,{\epsilon }_1={\epsilon }_2=0.15,\Delta \mu =0.6,U=0$ (bottom two overlapping lines) and $U=0.1$ (top two overlapping lines) $N_s=6$ and $\delta t=0.6$. The inset zooms on the time evolution after the early transients.

Figure 5

Model I with a vertical mirror symmetry. (a) Total current and its breakup into (b) $\mathcal{D}$ and (c)–(d) $\mathcal{R}$ components; $\varphi =\pi /2,{\gamma }_{\nu ,n}=0.1,{\epsilon }_1=0.1,{\epsilon }_2=0.2,\Delta \mu =0.6,U=0.1,N_s=6$, and $\delta t=0.6$.

Figure 6

Example of a convergence analysis for model I with a vertical mirror symmetry using the data of Fig. 5. (a) and (b) $\mathcal{R}(t,\pm \pi /2)$ with $\delta t=0.6$ and $N_s=2,3,...,6$. Panel (b) zooms on the long-time behavior. (c) and (d) Long-time $\left(t\sim 180\right)$ quasi-steady-state values of $\mathcal{R}(\pm \pi /2)$ and $\mathcal{D}(\pm \pi /2)$ as a function of ${\tau }_c=N_s\delta t$ for different time steps, $\delta t=0.6$ $(\circ )$, $\delta t=1$ $(\square )$, and $\delta t=1.2$ (triangle). Filled (empty) symbols stand for the $\varphi =\pi /2$ $\left(\varphi =-\pi /2\right)$ magnetic phases, overlapping in panel (d).

Figure 7

Model I. (a) Total current and its breakup into its (b) $\mathcal{R}$ and (c) $\mathcal{D}$ components for noncentrosymmetric and nondegenerate double-dot system; $\varphi =\pi /2,{\gamma }_{L,1}={\gamma }_{L,2}=0.2,{\gamma }_{R,1}={\gamma }_{R,2}=0.05,{\epsilon }_1=0.1,{\epsilon }_2=0.2,\Delta \mu =0.6,U=0.1,N_s=6$, and $\delta t=0.6$.

Figure 8

Steady-state behavior of model I with INFPI simulations using nondegenerate dots, ${\epsilon }_1=0.1,{\epsilon }_2=0.2$. Setups with a vertical mirror symmetry, ${\gamma }_{\nu ,n}=0.1$ $(\circ )$, and missing mirror symmetries, ${\gamma }_{L,n}=0.2$ and ${\gamma }_{R,n}=0.05$ $(\square )$. Other parameters are $\Delta \mu =0.6,U=0.1,\beta =50,N_s=6$, and $\delta t=0.6$.

Figure 9

Steady-state behavior of model I using a mean-field scheme, and the same parameters as in Fig. 8. Setups with a vertical mirror symmetry, ${\gamma }_{\nu ,n}=0.1$ $(\circ )$, and missing mirror symmetries, ${\gamma }_{L,n}=0.2$ and ${\gamma }_{R,n}=0.05$ $(\square )$.

Figure 10

Steady-state behavior of model I in the Coulomb blockade regime with the parameters of Fig. 8 using Eq. (18). Setups with a vertical mirror symmetry, ${\gamma }_{\nu ,n}=0.1$ $(\circ )$, and missing mirror symmetries, ${\gamma }_{L,n}=0.2$ and ${\gamma }_{R,n}=0.05$ $(\square )$.

Figure 11

Model II, an interferometer coupled to an equilibrium FE, with a vertical mirror symmetry. (a) Charge current in the AB interferometer, $U=0$ with $\varphi =\pm \pi /2$ (dot and circles, overlapping) and $U=0.1$ with $\varphi =\pi /2$ (dashed-dotted), $\varphi =-\pi /2$ (dashed). (b) and (c) Odd and even conductance terms obey the symmetries (16). The quantum dots in the AB interferometer are set at ${\epsilon }_{1,2}=0.15,{\gamma }_{\nu ,n}=0.05$. The FE is set at equilibrium $\left({\mu }_{\pm }=0\right)$ with ${\epsilon }_p=-0.5$ and ${\gamma }_{\pm }=0.2$. The four reservoirs are prepared at the temperature $1/\beta =50$. Numerical parameters are $\delta t=0.6,N_s=4$, and $L_s=120$.

Figure 12

Occupation of dot 1 in a system with a vertical mirror symmetry using the parameters of Fig. 11, $N_s=3,\delta t=0.8$.

Figure 13

Steady-state values of $\mathcal{R}$ and $\mathcal{D}$ in model II, symmetric ${\gamma }_{\nu ,n}=0.2$ $(\circ )$, and asymmetric ${\gamma }_{L,n}=0.05\ne {\gamma }_{R,n}=0.2$ $(\square )$ setups. Other parameters are the same as in Fig. 11.

Figure 14

Model II with the FE replaced by a voltage probe coupled to dot 1. The probe equations are detailed in Ref. [23]; symmetric ${\gamma }_{\nu ,n}=0.2$ $(\circ )$, and asymmetric ${\gamma }_{L,n}=0.05\ne {\gamma }_{R,n}=0.2$ $(\square )$ setups. Parameters are the same as in Figs. 11, 12, 13, besides the temperature which is set at $1/\beta =25$ and ${\gamma }_p=0.05$.

Figure 15

Model II with an out-of-equilibrium FE. (a) A finite drag current is induced in the interferometer (set in equilibrium $\Delta \mu =0)$, a result of its coupling to a nonequilibrium FE with $\Delta {\mu }_F=1$. Panel (b) demonstrates that the current in the interferometer is missing a linear response contribution (and all other odd powers of voltage); ${\epsilon }_{1,2}=0,{\gamma }_{\nu ,n}=0.1,{\gamma }_{\pm }=0.2,{\epsilon }_p=0.1,U=0.05,\beta =10$ (facilitating convergence), $\delta t=0.8$, and $N_s=3$.