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 an asymmetry is introduced, as we show numerically, but odd conductance terms only manifest a weak breakdown of the magnetic field symmetry. Under elastic dephasing effects, the Onsager-Casimir symmetry is maintained beyond linear response and under spatial asymmetries.