Perfect Diode in Quantum Spin Chains

We study the rectification of the spin current in $XXZ$ chains segmented in two parts, each with a different anisotropy parameter. Using exact diagonalization and a matrix product state algorithm, we find that a large rectification (of the order of $10^4$) is attainable even using a short chain of $N=8$ spins, when one-half of the chain is gapless while the other has a large enough anisotropy. We present evidence of diffusive transport when the current is driven in one direction and of a transition to an insulating behavior of the system when driven in the opposite direction, leading to a perfect diode in the thermodynamic limit. The above results are explained in terms of matching of the spectrum of magnon excitations between the two halves of the chain.

Figure 1

(a)–(d) We show the forward current ${\mathcal{J}}_f$ (a), the reverse current ${\mathcal{J}}_r$ (b), the rectification coefficient $\mathcal{R}$ (c), and the contrast $\mathcal{C}$ (d) as functions of the anisotropy ${\mathrm{\Delta }}_L/J_L$ for different chain lengths $N$. The current in reverse bias is significantly lower than in forward bias, resulting in large rectification and contrast. (e),(f) Rectification $\mathcal{R}$ (e) and contrast $\mathcal{C}$ (f) as functions of the system size for different values of the anisotropy ${\mathrm{\Delta }}_L/J_L$. The rectification $\mathcal{R}$ increases significantly with the system size for large enough anisotropy ${\mathrm{\Delta }}_L/J_L$. The other parameter values are ${\mathrm{\Delta }}_R={\mathrm{\Delta }}_M=0$, $J_L=J_R$, $J_M=J_L$, and $\gamma =J_L/\hslash$ for (a)–(d), while $J_M=0.1J_L$ and $\gamma =0.1J_L/\hslash$ for (e) and (f). The black dashed lines in (c) and (d) indicate ${\mathrm{\Delta }}_L/J_L=1+\sqrt{2}$.

Figure 2

Profile of the magnetization $⟨{\widehat{\sigma }}_n^z⟩$ as a function of the spin number $n$ for a chain of $N=20$ spins and for different values of the anisotropy ${\mathrm{\Delta }}_L/J_L$: (a) forward bias (${\lambda }_1=0.5$ and ${\lambda }_N=0$) and (b) reverse bias (${\lambda }_1=0$ and ${\lambda }_N=0.5$). In the bottom panels, we show the spin current (c) and the magnetization difference $\delta {\sigma }^z$ (d) as a function of the system size in the reverse-bias case. The other parameters values are as in Fig. 1.

Figure 3

(a) Contrast $\mathcal{C}$ versus anisotropy ${\mathrm{\Delta }}_L/J_L$. (b) Eigenfrequencies $E_{\nu ,R}$ and $E_{\nu ,L}$ as a function of ${\mathrm{\Delta }}_L/J_L$ computed from Eq. (6) for the right half (blue dashed lines) and left half of the chain (red continuous lines). The black dashed lines show that the dips in the contrast $\mathcal{C}$ in (a) occur when $E_{\nu ,R}+E_{\nu ,L}=0$. Parameter values: $N=10$, $J_M=J_L/10$, $\gamma =J_L/(10\hslash )$, and $J_R=J_L$.

Figure 4

(a) Contrast $\mathcal{C}$ versus $\delta \lambda$, which is the distance from ${\lambda }_n=0$, while the other bath parameter is kept at ${\lambda }_{n^{\prime }}=0.5$ (red continuous line) or the distance from ${\lambda }_n=0.5$ while the other bath parameter is kept at ${\lambda }_{n^{\prime }}=0$ (blue dashed line). Parameters are $N=10$, ${\mathrm{\Delta }}_R={\mathrm{\Delta }}_M=0$, ${\mathrm{\Delta }}_L/J_L=3$, $\gamma =J_R/\hslash$, and $J_M=J_L=J_R$. (b) Contrast $\mathcal{C}$ as a function of both ${\mathrm{\Delta }}_L/J_L$ and ${\mathrm{\Delta }}_R/J_R$. Parameters are $N=6$, $J_M=J_L=J_R$, and $\gamma =J_R/\hslash$.