Microscopic theory of heat transfer between two fermionic thermal baths mediated by a spin system

In this paper we have presented the heat exchange between the two fermionic thermal reservoirs which are connected by a fermionic system. We have calculated the heat flux using solution of the $c$-number Langevin equation for the system. Assuming small temperature difference between the baths we have defined the thermal conductivity for the process. It first increases as a nonlinear function of average temperature of the baths to a critical value then decreases to a very low value such that the heat flux almost becomes zero. There is a critical temperature for the fermionic case at which the thermal conductivity is maximum for the given coupling strength and the width of the frequency distribution of bath modes. The critical temperature grows if these quantities become larger. It is a sharp contrast to the Bosonic case where the thermal conductivity monotonically increases to the limiting value. The change of the conductivity with increase in width of the frequency distribution of the bath modes is significant at the low temperature regime for the fermionic case. It is highly contrasting to the Bosonic case where the signature of the enhancement is very prominent at high temperature limit. We have also observed that thermal conductivity monotonically increases as a function of damping strength to the limiting value at the asymptotic limit. There is a crossover between the high and the low temperature results in the variation of the thermal conductivity as a function of the damping strength for the fermionic case. Thus it is apparent here that even at relatively high temperature, the fermionic bath may be an effective one for the strong coupling between system and reservoir. Another interesting observation is that at the low temperature limit, the temperature dependence of the heat flux is the same as the Stefan-Boltzmann law. This is similar to the bosonic case.

Figure 1

Variation of heat conductance with average temperature $\left(T\right)$ of baths for the parameter set $\hslash =1.0,k_B=1.0,{\omega }_0=0.01$. (a) Fermionic baths, (b) bosonic baths (units are arbitrary).

Figure 2

(a) Plot of massless average fermion occupation number vs bath temperature for fermionic bath. (b) Plot of the rate of change of massless average fermion occupation number with bath temperature vs bath temperature for fermionic bath. (c) Plot of average phonon occupation number vs bath temperature for bosonic bath. (d) Plot of the rate of change of average phonon occupation number number with bath temperature vs bath temperature for bosonic bath. In each case, the parameter set is $\hslash =1.0,k_B=1.0$ (units are arbitrary).

Figure 3

Variation of heat conductance with Debye cutoff frequency $\left(D\right)$ for the parameter set $\hslash =1.0,k_B=1.0,{\omega }_0=0.01$. (a) Fermionic baths, (b) bosonic baths (units are arbitrary).

Figure 4

Variation of heat conductance with damping constant $\left({\gamma }_0\right)$ for fermionic bath for the parameter set $\hslash =1.0,k_B=1.0,{\omega }_0=0.01$. (a) fermionic baths, (b) bosonic baths (units are arbitrary).