Figure 1

(Color online) A schematic representation of the model with size $N=8$. The spin chain is connected to two phonon baths with coupling ${\lambda }_L$ and ${\lambda }_R$. The phonon baths are at different temperature $T_L$ and $T_R$. A magnetic field is applied to the spin chain.Reuse & Permissions

Figure 2

(Color online) Rectification as a function of the magnetic field in a chain of size $N=6$. The temperature of the heat bath are $T_{+}=T_0\left(1+\Delta \right)$ and $T_{-}=T_0\left(1-\Delta \right)$, where $T_0=0.1$ is the mean temperature. The coupling between the spin chain and heat bath are ${\lambda }_L=0.16$ and ${\lambda }_R=0.04$, anisotropy parameter $\gamma =0.1$. The square, circle, and triangular correspond to $\Delta =0.2$, 0.4, and 0.8, respectively.Reuse & Permissions

Figure 3

(Color online) Analysis of the reversal of rectification in a two-spin system. Parameters: $T_0=1.0$, $\Delta =0.8$, ${\lambda }_L=0.16$, ${\lambda }_R=0.04$, and $\gamma =0.1$. (a) The eigenvalues $E_n$ vs magnetic field $h$. (b) The density ${\rho }_{nn}$ (probability of each eigenstate) vs $h$. (c) The product of $E_n$ and the derivative of ${\rho }_{nn}$ from high-temperature heat bath, which is the contribution to heat flux from each eigenstate. In (a), (b), and (c), the square, circle, up triangle, and down triangle correspond to eigenstates 1, 2, 3, and 4, respectively. In (b) and (c), the solid and hollow symbols correspond to forward and backward thermal transport, respectively. (d) The heat fluxes $J_{+}$ and $J_{-}$ vs magnetic field $h$ (left scale). The square curve shows rectification effect $R$ vs magnetic field $h$ (right scale).Reuse & Permissions

Figure 4

(Color online) The rectification $R$ vs nonuniform magnetic field. The chain size is $N=6$, $h\left(1:N/2\right)$ is the magnetic field applied to site 1 to $N/2$, and others are zero, that is, $h\left(N/2+1:N\right)=0$. Here $\gamma =0$, $T_0=0.1$, and ${\lambda }_L={\lambda }_R=0.1$. The square, circle and triangular correspond to $\Delta =0.2$, 0.4, and 0.8, respectively.Reuse & Permissions

Figure 5

(Color online) Rectification as a function of mean temperature $T_0$. The size of spin chain is $N=6$. The temperature of the heat bath are $T_{+}=T_0+\Delta T$ and $T_{-}=T_0-\Delta T$. Here, ${\lambda }_L=0.16$, ${\lambda }_R=0.04$, $\gamma =0.1$, and $h=0.01$. The square, circle, and triangular correspond to $\Delta T=0.01$, 0.02, and 0.04, respectively.Reuse & Permissions

Figure 6

(Color online) The rectification changes with magnetic field for the spin chain with different sizes. The curves of square, circle, and up triangle, and down triangle, star, and diamond correspond to $N=2$, 4, 6, 8, 10, and 12, respectively. Here the parameters are: $T_{+}=0.18$, $T_{-}=0.02$, ${\lambda }_L=0.16$, ${\lambda }_R=0.04$, and $\gamma =0.1$.Reuse & Permissions

Figure 7

(Color online) The rectification changes with anisotropy in the spin chain. The chain size is $N=6$. The square, circle, and triangular correspond to $\Delta =0.2$, 0.4, and 0.8, respectively. Here the parameters are: $T_{+}=0.18$, $T_{-}=0.02$, ${\lambda }_L=0.16$, ${\lambda }_R=0.04$, and $h=0.01$.Reuse & Permissions