# Giant Spin Current Rectification Due to the Interplay of Negative Differential Conductance and a Non-Uniform Magnetic Field

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## Abstract

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## 1. Introduction

## 2. Model

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NDC | Negative Differential Conductivity |

GKSL | Gorini-Kossakowski-Sudarshan-Lindblad |

NESS | Non-equilibrium steady state |

## References

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**Figure 1.**Spin current $\mathcal{J}$ as a function of the ratio of interaction and local field strength ${J}_{zz}/h$ for system sizes $L=4$ (

**a**), $L=6$ (

**b**) and $L=8$ (

**c**). Different lines corresponds to each of the ${2}^{L}$ magnetic field configurations. We highlight two magnetic field configurations: with the red dotted line, we show the current for a field, which is h for the first half of the chain, and $-h$ for the second half of the chain, which we refer to as $(+,\cdots ,+,-,\cdots ,-)$, and with the blue dashed line the realization in which the field is $-h$ in the first half of the chain and h in the second half $(-,\cdots ,-,+,\cdots ,+)$. The common parameters are $h=4$, $\gamma =1$ and $\mu =1$.

**Figure 2.**Spin current $\mathcal{J}$ as a function of the ratio of local field strength and interaction $h/{J}_{zz}$ for system sizes $L=4$ (

**a**), $L=6$ (

**b**) and $L=8$ (

**c**). Different lines corresponds to each of the ${2}^{L}$ configurations of local fields. We highlight two magnetic field configurations: with the red dotted line we show the current for a field, which is h for the first half of the chain, and $-h$ for the second half of the chain, which we refer to as $(+,\cdots ,+,-,\cdots ,-)$, and with the blue dashed line the realization in which the field is $-h$ in the first half of the chain and h in the second half $(-,\cdots ,-,+,\cdots ,+)$. Peaks of red dotted line in panel (

**a**) are signalled by black dashed lines that correspond to the black dashed lines in Figure 5. Common parameters are ${J}_{zz}=4$, $\gamma =1$ and $\mu =1$.

**Figure 3.**Rectification $\mathcal{R}$ is plotted as a function of the ratio of local field strength and interaction $h/{J}_{zz}$ for system sizes $L=4$ (

**a**), $L=6$ (

**b**) and $L=8$ (

**c**). The current ${\mathcal{J}}_{f}$ with magnetic field configuration as $(+,\cdots ,+,-,\cdots ,-)$ and ${\mathcal{J}}_{r}$ for $(-,\cdots ,-,+,\cdots ,+)$ are highlighted as thick blue lines. The other configurations are in thin grey lines. Common parameters are ${J}_{zz}=4$, $\gamma =1$, $\mu =1$.

**Figure 4.**Rectification $\mathcal{R}$ is plotted as a function of interaction ${J}_{zz}$ for $h=0.1$ (

**a**), $h=1$ (

**b**) and $h=3$ (

**c**) for a system size of $L=8$. The rectification as ratio of current ${\mathcal{J}}_{f}$ with magnetic field configuration as $(+,\cdots ,+,-,\cdots ,-)$ and ${\mathcal{J}}_{r}$ for $(-,\cdots ,-,+,\cdots ,+)$ are highlighted as thick blue lines. The other configurations are in thin grey lines. The common parameters are $\gamma =1$, $\mu =1$.

**Figure 5.**Eigenenergies ${E}_{s}$ in the zero magnetisation sector of a chain of length $L=4$ with magnetic field configuration $(+,+,-,-)$ plotted as a function of the ratio of local field strength and interaction $h/{J}_{zz}$. The vertical lines in each panel correspond to peaks in the current as from Figure 2a. The color that is used for the eigenenergies corresponds to the overlap between the eigenvector and the state $|\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\downarrow \downarrow \uparrow \uparrow \rangle $. Parameters: $\gamma =1$, $\mu =1$, ${J}_{zz}=4$.

**Figure 6.**The inverse participation ratio $1-IPR$ as a function of the ratio of local field strength and interaction $h/{J}_{zz}$ for different system sizes $L=16$ (dashed line), $L=10$ (continuous line) and $L=6$ (dot-dashed line). Here the magnetic field is in the configuration $(+,\cdots ,+,-,\cdots ,-)$. Both state configurations $|DU\rangle =|\phantom{\rule{-0.166667em}{0ex}}\downarrow \cdots \downarrow \uparrow \cdots \uparrow \rangle $ (red) and $|UD\rangle =|\phantom{\rule{-0.166667em}{0ex}}\uparrow \cdots \uparrow \downarrow \cdots \downarrow \rangle $ (blue) are shown. The inset shows the rectification as a function of $h/{J}_{zz}$ for different system sizes: blue continuous line for $L=4$, red dashed line for $L=6$, and green dot-dashed line for $L=8$. Other parameters are ${J}_{zz}=4$, $\gamma =1$, $\mu =1$.

**Figure 7.**The Von Neumann entropy $\mathcal{S}$ as a function of the ratio of local field strength and interaction $h/{J}_{zz}$ for system sizes $L=4$ (blue solid line), $L=6$ (red dashed line) and $L=8$ (green dot-dashed line). We show both the entropy of the steady state density matrix ${\rho}_{ss}$ for the magnetic field configuration $(+,\cdots ,+,-,\cdots ,-)$ (no symbols) and $(-,\cdots ,-,+,\cdots ,+)$ (lines with ∘). The common parameters are ${J}_{zz}=4$, $\gamma =1$ and $\mu =1$.

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**MDPI and ACS Style**

Lee, K.H.; Balachandran, V.; Tan, R.; Guo, C.; Poletti, D.
Giant Spin Current Rectification Due to the Interplay of Negative Differential Conductance and a Non-Uniform Magnetic Field. *Entropy* **2020**, *22*, 1311.
https://doi.org/10.3390/e22111311

**AMA Style**

Lee KH, Balachandran V, Tan R, Guo C, Poletti D.
Giant Spin Current Rectification Due to the Interplay of Negative Differential Conductance and a Non-Uniform Magnetic Field. *Entropy*. 2020; 22(11):1311.
https://doi.org/10.3390/e22111311

**Chicago/Turabian Style**

Lee, Kang Hao, Vinitha Balachandran, Ryan Tan, Chu Guo, and Dario Poletti.
2020. "Giant Spin Current Rectification Due to the Interplay of Negative Differential Conductance and a Non-Uniform Magnetic Field" *Entropy* 22, no. 11: 1311.
https://doi.org/10.3390/e22111311