Magnetic field symmetries of nonlinear transport with elastic and inelastic scattering

We study nonlinear electronic transport symmetries in Aharonov-Bohm interferometers subjected to inelastic-scattering effects and show that odd (even) conductance terms are even (odd) in the magnetic field when the junction is (left-right) spatially symmetric. This observation does not hold when spatial inversion symmetry is broken, as we show numerically. Under elastic dephasing effects, the Onsager-Casimir symmetry is maintained beyond linear response, irrespective of spatial asymmetries.

Figure 1

Scheme of our setup. The horizontal arrow represents the charge current $I$. The two parallel arrows represent (same magnitude) currents into and from the $P$ terminal, serving to induce elastic- and or inelastic-scattering effects.

Figure 2

(a) Spatially symmetric junctions with ${\gamma }_L={\gamma }_R=0.05$ and $\mathcal{R}=\Delta I$. (b), (c) Spatial asymmetry is introduced, with ${\gamma }_L=0.05$, ${\gamma }_R=0.2$, resulting in $\mathcal{R}\ne \Delta I$. We adopt a voltage probe mimicking inelastic effects with ${\gamma }_P=0.1$, ${\epsilon }_1={\epsilon }_2=0.15$, and inverse temperature $1/T_{\nu }=50$. In all plots $\mathcal{R}$ is represented by $\circ$ and $\Delta I$ is represented by dashed lines.

Figure 3

Probe chemical potential (a) in spatially symmetric junctions ${\gamma }_L={\gamma }_R=0.05$ and (b) when spatial asymmetry is introduced, ${\gamma }_L=0.05$ and ${\gamma }_R=0.2$. In all plots $\varphi =\pi /4$ (dot) and $\varphi =-\pi /4$ ($\circ$), the light dotted line presents the symmetry line, and the insets zoom on the main plots. Other parameters are the same as in Fig. 2

Figure 4

(a) Even ($\mathcal{R}$) and (b) odd ($\mathcal{D}$) conductance terms in spatially symmetric junctions, ${\gamma }_L={\gamma }_R=0.05$ (full) and with an asymmetry ${\gamma }_L=0.05\ne {\gamma }_R=0.2$ (dotted) for ${\epsilon }_1={\epsilon }_2=0.15$. (c), (d) Same as above, only with nondegenerate levels ${\epsilon }_1=0.1$ and ${\epsilon }_2=0.2$. Parameters are the same as in Fig. 2, $\Delta \mu =0.4$. Light dotted lines present the symmetry lines.