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 $T=0$ and the dynamical Onsager matrix is fully symmetric, 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 nonequilibrium 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.

Figure 1

Sketch of the setup. A quantum dot connected to a single electronic reservoir at $T=0$ is driven by means of a gate voltage $V_g\left(t\right)$ and a magnetic field $B\left(t\right)$. We model the device by an Anderson Hamiltonian with a single level and Coulomb interaction $U$.

Figure 2

Generalized KS relations computed by NRG for $U=0.03$ and $\mathrm{\Delta }=0.0008$. The driving protocol is $V\left(t\right)=V_{0}sin\left(\mathrm{\Omega }t\right),B\left(t\right)=B_{0}sin(\mathrm{\Omega }t+\pi /4)+0.0005$ with $V_0=0.006$, and $B_0=0.0003$. Energies are expressed in units of the bandwidth of the reservoir: (top) Eq. (30), (middle) Eq. (31), and (bottom) Eq. (32). Solid lines correspond to the direct calculation of the Onsager coefficients ${\mathrm{\Lambda }}_C$ and symbols to the evaluations from the frozen occupancies.

Figure 3

Sketch of the evolution of the occupancy of the QD. The energy of the singly and doubly occupied states have an energy gap $\delta$ which depends on the Coulomb interaction and on the magnetic field.

Figure 4

Occupancy map of $n_{f,\downarrow }$ in the mean field approximation for $U=0.01$ and $\mathrm{\Delta }=0.0008$. The green ellipse indicates the driving protocol $V\left(t\right)=V_{0}sin\left(\mathrm{\Omega }t\right),B\left(t\right)=B_{0}sin(\mathrm{\Omega }t+\pi /4)+0.0005$ with $V_0=0.02$ and $B_0=0.001$. Left and right panels correspond to results computed with mean field and NRG, respectively. Energies are expressed in units of the bandwidth $D$ of the reservoir.

Figure 5

Frozen occupation along the curve of Fig. 4. Solid line (blue: $n_{f\downarrow }\left(t\right)$ computed by NRG. Dashed line (black) $n_{f\downarrow }\left(t\right)$ computed in mean field approximation for parameters along the green curve of Fig. 4. Circles (red) $n_{f\uparrow }\left(t\right)$ computed by NRG.

Figure 6

Computed powers with NRG. $U=0.03$ and $\mathrm{\Delta }=0.0008$, under the driving protocol $V_g\left(t\right)=V_{0}sin\left(\mathrm{\Omega }t\right)$, $B\left(t\right)=B_{0}sin(\mathrm{\Omega }t+\pi /4)+0.0005)$ with $V_0=0.006$ and $B_0=0.0003$. Energies are expressed in units of the bandwidth of the reservoir. (Top) Plots in solid red, dashed blue, green circles, and black stars correspond, respectively, to $P_{\uparrow }\left(t\right)$, $P_{\downarrow }\left(t\right)$, $P_{\mathrm{Joule},\uparrow }\left(t\right)$, and $P_{\mathrm{Joule},\downarrow }\left(t\right)$. (Middle) Plots in solid line red and dashed blue correspond to $P_{\uparrow }\left(t\right)-P_{\text{Joule},\uparrow }\left(t\right)$ and $P_{\downarrow }\left(t\right)-P_{\text{Joule},\downarrow }\left(t\right)$, respectively. (Bottom) The total power $P_{\uparrow }\left(t\right)+P_{\downarrow }\left(t\right)$ is plotted in solid black and coincides with the total total Joule dissipation. $P_{\mathrm{Joule},\uparrow }\left(t\right)+P_{\mathrm{Joule},\downarrow }\left(t\right)$, which is plotted in red circles.

Figure 7

Minimal eigenvalue (in log scale) of the matrix $\mathrm{\Lambda }\left(\mathbf{V}\right)$ for $U=0.01$ and $\mathrm{\Delta }=0.0008$. Energies are expressed in units of the bandwidth of the reservoir.

Figure 8

Mean charge accumulation for each spin orientation for $U=0.03$ and $\mathrm{\Delta }=0.0008$, computed with NRG, as a function of the phase difference $\varphi$ of the driving protocol $V_g\left(t\right)=V_{0}sin\left(\mathrm{\Omega }t\right),B\left(t\right)=B_{0}sin(\mathrm{\Omega }t+\varphi )+0.0005)$ for $V_0=0.006$ and $B_0=0.0003$. Dashed line (blue) $\sigma =\downarrow$. Solid line (red) $\sigma =\uparrow$.