Flux rectification in the quantum $XXZ$ chain

Thermal rectification is the phenomenon by which the flux of heat depends on the direction of the flow. It has attracted much interest in recent years due to the possibility of devising thermal diodes. In this paper, we consider the rectification phenomenon in the quantum XXZ chain subject to an inhomogeneous field. The chain is driven out of equilibrium by the contact at its boundaries with two different reservoirs, leading to a constant flow of magnetization from one bath to the other. The nonunitary dynamics of this system, which is modeled by a Lindblad master equation, is treated exactly for small sizes and numerically for larger ones. The functional dependence of the rectification coefficient on the model parameters (anisotropy, field amplitude, and out of equilibrium driving strength) is investigated in full detail. Close to the XX point and at small inhomogeneity and low driving, we have found an explicit expression for the rectification coefficient that is valid at all system sizes. In particular, it shows that the phenomenon of rectification persists even in the thermodynamic limit. Finally, we prove that in the case of the XX chain, there is no rectification.

Figure 1

Flux $J$ vs $f$ for $N=3,\Delta =1$, and different values of $h$. The points correspond to simulations and the solid line corresponds to Eq. (34). Inset: Same but for $N=2$ [cf. Eq. (33)].

Figure 2

$J\left(f\right)$ (upper curves) and $J(-f)$ (lower curves) vs $\Delta$ for $N=3$ and $h=0$, 1, 2, and 4 (as signaled in each figure). (a) $f=0.5$, (b) $f\to 0$, and (c) $f=1$. In (b), we plot the limit of $J/f$ as $f\to 0$.

Figure 3

Rectification coefficient $R$ vs $\Delta$ for $N=3,f=0.5$, and different values of $h$, as signaled in the figure. See Eq. (35).

Figure 4

$J\left(f\right)$ (upper curve) and $J(-f)$ (lower curve) vs $\Delta$ for $f=0.5$, different values of $h$ (as shown in the label of each image), and different sizes [as shown in image (d)].

Figure 5

$R$ vs $\Delta$ for $f=0.5$, different values of $h$ (as shown in the label of each image), and different sizes. The color scheme is the same as Fig. 4.

Figure 6

The function $A_N=R/fh\Delta$ for different values of $N$, as obtained from arbitrary precision simulations. The solid line corresponds to Eq. (39).

Figure 7

The rectification coefficient $R/fh$ vs $\Delta$ for $f=0.01,h=0.01$, and different values of $N$, as denoted in the figure.