Thermodynamic meaning of local temperature of nonequilibrium open quantum systems

Measuring the local temperature of nanoscale systems out of equilibrium has emerged as a new tool to study local heating effects and other local thermal properties of systems driven by external fields. Although various experimental protocols and theoretical definitions have been proposed to determine the local temperature, the thermodynamic meaning of the measured or defined quantities remains unclear. By performing analytical and numerical analysis of bias-driven quantum dot systems both in the noninteracting and strongly-correlated regimes, we elucidate the underlying physical meaning of local temperature as determined by two definitions: the zero-current condition that is widely used but not measurable and the minimal-perturbation condition that is experimentally realizable. We show that, unlike the zero-current condition, the local temperature determined by the minimal-perturbation protocol establishes a quantitative correspondence between the nonequilibrium system of interest and a reference equilibrium system, provided the probed system observable and the related electronic excitations are fully local. The quantitative correspondence thus allows the well-established thermodynamic concept to be extended to nonequilibrium situations.

Figure 1

Schematic illustration of Eq. (20). The local observable $O_0$ of a nonequilibrium QD can be made equivalent to that of a reference equilibrium QD, provided the two dots have the same local temperature $T^{*}$.

Figure 2

Calculated (a) $T^{*}$ and (b) relative deviation between ${\chi }_0^m(T,V)$ and ${\chi }_0^m(T^{*},0)$ versus ${\epsilon }_d$ for a noninteracting QD under a bias voltage $V$. (c) Dot spectral function $A\left(\omega \right)$ and (d) $\delta {\chi }_p^m$ versus $T_p$ for different ${\epsilon }_d$. The QD parameters are (in units of $\mathrm{\Delta }$): $U=0,T_L=T_R=T=0.1,{\mu }_R=-{\mu }_L=V/2=0.2,{\mathrm{\Delta }}_L={\mathrm{\Delta }}_R=0.5,{\mathrm{\Omega }}_L={\mathrm{\Omega }}_R=0$, and $W_L=W_R=20$. The vertical lines and regions are explained in the main text.

Figure 3

Calculated $T^{*}$ versus $U$ for an interacting QD under a bias voltage $V$ at the background temperature (a) $T=\mathrm{\Delta }$ and (b) $T=0.1\mathrm{\Delta }$. The relative deviations between $O_0(T,V)$ and $O_0(T^{*},0)$ are shown in (c) and (d), respectively. The QD parameters are (in units of $\mathrm{\Delta }$): ${\epsilon }_d=-2,{\mu }_R=-{\mu }_L=V/2=0.2,{\mathrm{\Delta }}_L={\mathrm{\Delta }}_R=0.5,{\mathrm{\Omega }}_L={\mathrm{\Omega }}_R=0$, and $W_L=W_R=20$. The vertical lines and regions are explained in the main text.

Figure 4

Dot spectral function $A\left(\omega \right)$ of a nonequilibrium QD with a varying of $U$ at the background temperature (a) $T=\mathrm{\Delta }$ and (b) $T=0.1\mathrm{\Delta }$. The inset of (b) shows the Kondo spectral peak at various $T$. The vertical lines mark the excitation energy window $({\mu }_L-{\omega }_L,{\mu }_R+{\omega }_R)$. The QD parameters are the same as in the caption of Fig. 3.