Work exchange, geometric magnetization and fluctuation-dissipation relations in a quantum dot under adiabatic magnetoelectric driving

We study the adiabatic dynamics of the charge, spin and energy of a quantum dot with a Coulomb interaction under two-parameter driving, associated to time-dependent gate voltage and magnetic field. The quantum dot is coupled to a single reservoir at temperature $T=0$, hence, the net energy dynamics is fully dissipative. However, in the presence of many-body interactions, other interesting mechanisms take place, like the net exchange of work between the two types of forces and the non-equilibrium accumulation of charge with different spin orientations. The latter has a geometric nature. The dissipation takes place in the form of an instantaneous Joule law with the universal resistance $R_0=h/2e^2$. We show the relation between this Joule law and instantaneous fluctuation-dissipation relations. The latter lead to generalized Korringa-Shiba relations, valid in the Fermi-liquid regime.