Heat transfer in the spin-boson model: A comparative study in the incoherent tunneling regime

We study the transfer of heat in the nonequilibrium spin-boson model with an Ohmic dissipation. In the nonadiabatic limit we derive a formula for the thermal conductance based on a rate equation formalism at the level of the noninteracting blip approximation, valid for temperatures $T>T_K$, with $T_K$ as the Kondo temperature. We evaluate this expression analytically assuming either weak or strong couplings, and demonstrate that our results agree with exact relations. Far-from-equilibrium situations are further examined, showing a close correspondence to the linear response limit.

Figure 1

The Kondo Temperature $T_K$ of Eq. (8) with ${\alpha }_L={\alpha }_R$, ${\omega }_c=10\Delta$. NIBA results are valid in the high temperature regime.

Figure 2

Thermal conductance as a function of $\alpha$, calculated from Eq. (19) with the nonadiabatic rates (16), assuming an Ohmic form and ${\alpha }_L={\alpha }_R$, ${\omega }_c=10\Delta$.

Figure 3

Thermal conductance as a function of $\alpha$, with the same data as in Fig. 2. (a) Weak coupling limit: NIBA expression Eq. (19) (symbols), Born-Markov result (9) (dashed lines). (b) Strong coupling limit, demonstrating the scaling (21). The legend describes both panels.

Figure 4

The NESB model beyond linear response: Heat current (15) divided by the temperature difference and thermal conductance, Eq. (19) at (a) ${\alpha }_R={\alpha }_L$, (b) ${\alpha }_L\ne {\alpha }_R$. In both cases $k_B{T}_L/\hslash \Delta =1.5$, $k_B{T}_R/\hslash \Delta =0.5$, ${\omega }_c=10\Delta$. The thermal conductance is evaluated at the average temperature $T=(T_L+T_R)/2$.