Modeling heat transport through completely positive maps

We investigate heat transport in a spin-$\frac{1}{2}$ Heisenberg chain, coupled locally to independent thermal baths of different temperature. The analysis is carried out within the framework of the theory of open systems by means of appropriate quantum master equations. The standard microscopic derivation of the weak-coupling Lindblad equation in the secular approximation is considered, and shown to be inadequate for the description of stationary nonequilibrium properties like a nonvanishing energy current. Furthermore, we derive an alternative master equation that is capable of describing a stationary energy current and, at the same time, leads to a completely positive dynamical map. This paves the way for efficient numerical investigations of heat transport in larger systems based on Monte Carlo wave function techniques.

Figure 1

Spin-$\frac{1}{2}$ chain coupled to heat baths of different temperature. The stationary state features a nonvanishing energy flux through the chain of spins from the hotter toward the colder heat bath.

Figure 2

Energy current $⟨{\widehat{J}}^{\left(1,2\right)}⟩$ in a chain of $n=3$ weakly coupled spins, driven by heat baths of different temperature $\left({\beta }_L=0.41,{\beta }_R=1.39,\lambda =\kappa =0.01,\Omega =1\right)$ Solid line: Computed on the basis of the Redfield master Eq. (26) by exact diagonalization in Liouville space. Filled circles: Weak internal coupling approximation (62) (again exact diagonalization). Circles: Numerical solution of Eq. (62) by means of a stochastic wave function simulation (${10}^5$ realizations).