Local temperatures and heat flow in quantum driven systems

We discuss the concept of local temperature for quantum systems driven out of equilibrium by ac pumps showing explicitly that it is the correct indicator for heat flow. We also show that its use allows for a generalization of the Wiedemann-Franz law.

Figure 1

Scheme of the setup. The central device is a wire connected to the left and right reservoirs. The third reservoir (P) represents the thermometer, which consists of a macroscopic system weakly coupled to a given point of the central device.

Figure 2

Local temperature along a one-dimensional model of $N=50$ sites with two ac fields operating with a phase lag of $\delta =\pi /2$ at the positions indicated by dotted lines. The system is in contact with reservoirs with chemical potentials $\mu =0.2$ and temperature $T=0.025$. The driving frequency is ${\Omega }_0=0.05$ and the amplitude is $V_0=0.05$.

Figure 3

dc heat current divided by $V_0^2$ (solid) and local temperature difference (dashed) between the system and the left reservoir (blue) or the right reservoir (red) as a function of driving frequency ${\Omega }_0$. The phase lag is $\delta =1.88$ and the driving amplitude is $V_0=0.05$. The temperature and the chemical potential of the reservoirs are $T=0.025$ and $\mu =0.2$.

Figure 4

Upper panel: Heat flow (solid) and local temperature difference (dashed) between the system and the left reservoir (blue) or the right reservoir (red) as a function of the temperature $T$ of the reservoirs. The phase lag is $\delta =\pi /2$, the driving frequency is ${\Omega }_0=0.01$ and the amplitude is $V_0=0.05$. The chemical potential of the reservoirs is $\mu =0.2$. Lower panel: Heat flow (solid) and local temperature difference (dashed) between the system and the left reservoir (blue) or the right reservoir (red) as a function of the phase lag. The driving frequency is ${\Omega }_0=0.01$ and the amplitude is $V_0=0.05$. The temperature and the chemical potential of the reservoirs are $T=0.025$ and $\mu =0.2$.

Figure 5

$K/G$ for the right contact (red crosses) as a function of temperature. The black dashed line represents the Wiedemann-Franz law. Inset: The black dotted line represents the behavior of the quotient $K/G$ (red line) for very low temperatures, as depicted in Eq. (40). The phase-lag is $\delta =\pi /2$, the driving frequency is ${\Omega }_0=0.005$ and the amplitude is $V_0=0.05$. The chemical potential of the reservoirs is $\mu =0.2$.