Heat rectification on the $XX$ chain

In order to better understand the minimal ingredients for thermal rectification, we perform a detailed investigation of a simple spin chain, namely, the open $XX$ model with a Lindblad dynamics involving global dissipators. We use a Jordan-Wigner transformation to derive a mathematical formalism to compute the heat currents and other properties of the steady state. We have rigorous results to prove the occurrence of thermal rectification even for slightly asymmetrical chains. Interestingly, we describe cases where the rectification does not decay to zero as we increase the system size, that is, the rectification remains finite in the thermodynamic limit. We also describe some numerical results for more asymmetrical chains. The presence of thermal rectification in this simple model indicates that the phenomenon is of general occurrence in quantum spin systems.

Figure 1

Rectification profile for a junction of external magnetic fields (66) composed of 50 sites. The difference in temperature is given by $\mathrm{\Delta }T=T_L-T_R=5$ and the interaction is ${\alpha }_i=1$.

Figure 2

Rectification profile for a junction of external magnetic fields (66) composed of 50 sites. The difference in temperature is given by $\mathrm{\Delta }T=T_L-T_R=5$ and the interaction is ${\alpha }_i=5$.

Figure 3

Rectification profile for a junction of two interactions (67). The system is composed of 50 sites and the difference in temperature is given by $\mathrm{\Delta }T=T_L-T_R=5$, without a magnetic field.

Figure 4

Rectification profile for a junction of two interactions (67). The system is composed of 50 sites and the difference in temperature is given by $\mathrm{\Delta }T=T_L-T_R=5$. The magnetic field is fixed at $h_i=5$.

Figure 5

Rectification profile for a linear graded external magnetic field. The system is composed fo ten sites and the difference in temperature is given by $\mathrm{\Delta }T=5$, 10, and 15. The intersite interaction is fixed at ${\alpha }_i=1$. The external graded magnetic field is given by $h_i\propto ih$.

Figure 6

Rectification profile for a linear graded interaction. The system is composed of ten sites and the difference in temperature is given by $\mathrm{\Delta }T=5$, 10, and 15. The external magnetic field is fixed at $h_i=5$ and the intersite interaction is given by ${\alpha }_i\propto i\alpha$.

Figure 7

Rectification profile for a linear graded chain. The external magnetic field and the intersite interaction are linear on $\delta$, more specifically, $h_i\propto i\delta$ and ${\alpha }_i\propto i\delta$. The system is composed of ten sites and the difference in temperature is given by $\mathrm{\Delta }T=5$, 10, and 15.