Heat current rectification in segmented $XXZ$ chains

We study the rectification of heat current in an $XXZ$ chain segmented in two parts. We model the effect of the environment with Lindblad heat baths. We show that in our system, rectification is large for strong interactions in half of the chain and if one bath is at a cold enough temperature. For the numerically accessible chain lengths, we observe that the rectification increases with the system size. We gain insight into the rectification mechanism by studying two-time correlations in the steady state. The presence of interactions also induces a strong nonlinear response to the temperature difference, resulting in superlinear and negative differential conductance regimes.

Figure 1

Role of interaction. (a) Rectification coefficient $\mathcal{R}$ as a function of anisotropy parameter $\mathrm{\Delta }$ for different chain lengths $N$. Cold bath temperature is $T_C=0.1$ and hot bath temperature is $T_H=10+T_C$. (b) Dependence of rectification coefficient $\mathcal{R}$ on anisotropy parameter $\mathrm{\Delta }$ with different cold bath temperature $T_C$ for a chain of length $N=8$ and $T_H=10+T_C$. In both panels we used $J_{N/2}=1$.

Figure 2

Role of $\mathrm{\Delta }T$. (a) Variation of rectification coefficient $\mathcal{R}$ with temperature bias $\mathrm{\Delta }T$ for different anisotropy parameter $\mathrm{\Delta }$. Here, the cold bath temperature is $T_C=0.1$. (b) Dependence of rectification coefficient $\mathcal{R}$ on temperature bias $\mathrm{\Delta }T$ for different cold bath temperature $T_C$. Here the anisotropy $\mathrm{\Delta }=4$. In both panels, the chain length $N=8,T_H=T_C+\mathrm{\Delta }T$, and $J_{N/2}=1$.

Figure 3

Interface tunneling. Rectification ratio vs anisotropy $\mathrm{\Delta }$ for different magnitudes of the tunneling at the interface between the interacting and the noninteracting halves of the chain (purple circles for $J_{N/2}=0.1$, blue squares for $J_{N/2}=0.2$, red triangles for $J_{N/2}=0.5$, and green diamonds for $J_{N/2}=1.0$). The smaller the interface tunneling $J_{N/2}$, the stronger is the rectification effect, however the larger are the resonance effects due to finite size. Other parameters are $N=8,T_C=0.1$, and $T_H=10$.

Figure 4

Frequency response. $\mathrm{Im}\left[\mathcal{C}\right(\omega \left)\right]$, i.e., the imaginary part of the Fourier transform of the two-time correlation $C\left(t\right)$ in Eq. (11) as a function of the frequency $\omega$. We consider different values of the anisotropy (red diamonds for $\mathrm{\Delta }=1$, yellow circles for $\mathrm{\Delta }=2$, purple squares for $\mathrm{\Delta }=3$, green downward triangles for $\mathrm{\Delta }=4$, and blue upward triangles for $\mathrm{\Delta }=5$) for forward (a) and reverse bias (b). Other parameters are $N=8,T_C=1,T_H=100,\gamma =1$, and $J_{N/2}=1$.

Figure 5

Negative differential conductance. Reverse (a) and forward bias current (b) vs $T_C$ for different anisotropies $\mathrm{\Delta }=5$ (green triangles), $\mathrm{\Delta }=4$ (purple squares), $\mathrm{\Delta }=3$ (yellow circles), and $\mathrm{\Delta }=2$ (red diamonds). The hot bath temperature is set to $T_H=10$. Other parameters are $N=8,\gamma =1$, and $J_{N/2}=1$. In the inset of panel (a), the range for the $y$ axis is modified so that for $\mathrm{\Delta }=2$ the monotonic behavior of the heat current can be seen down to low values of $T_C$.

Figure 6

Scaling of rectification reversal. Variation of rectification coefficient $\mathcal{R}$ with temperature bias $\mathrm{\Delta }T$ for different chain lengths $N$. The dotted line corresponds to $\mathcal{R}=1.$ Here, the anisotropy is $\mathrm{\Delta }=5$, cold bath temperature $T_C=0.1$, hot bath temperature $T_H=T_C+\mathrm{\Delta }T$, and $J_{N/2}=1$.

Figure 7

Frequency response for a half-chain. $|{\mathcal{C}}_1\left(\omega \right)|$, i.e., the absolute value of the Fourier transform of the two-time correlation $C_1\left(t\right)$ as a function of the frequency $\omega$ with $T=1$, for different anisotropies $\mathrm{\Delta }$. The length of the chain is $N=4$ and $\gamma =1$.