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SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 65 , Issue 2 : 227212(2022) https://doi.org/10.1007/s11433-021-1805-6

Spin current Kondo effect in frustrated Kondo systems

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  • ReceivedSep 4, 2021
  • AcceptedOct 28, 2021
  • PublishedDec 29, 2021
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Abstract


Acknowledgment

This work was supported by the National Key Research and Development Program of China (Grant No. 2017YFA0303103), the National Natural Science Foundation of China (Grant Nos. 12174429, 11774401, and 11974397), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB33010100), and the Youth Innovation Promotion Association of Chinese Academy of Sciences.


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  • Figure 1

    (Color online) The three-impurity Kondo model (3IKM) and its phase diagram. (a) Schematic picture of 3IKM, showing three antiferromagnetically coupled impurity spins (red bold arrows) sitting on the vertices of an equilateral triangle with side length $R$ and Kondo coupled to a two-dimensional electron bath with a parabolic dispersion. (b) The large-$N$ phase diagram of 3IKM in terms of $R$ and $T_{\rm~K}/J_H$. The black curves with data points are the phase boundaries determined by the jump of holon phase shift, $\delta_\chi$. The number in each phase denotes the value $\delta_\chi/\pi$. The intensity plot with a red-blue color scale shows the correlation between the conduction electron spin current $\boldsymbol{j}_{ij}^s$ and the impurity spin vector chirality $\boldsymbol{J}_{ij}^S\equiv~{\boldsymbol~S}_i\times~{\boldsymbol~S}_j$. (c) The spin states of different phases for spin size $S=1/2$ ($K=1$) and $S=1$ ($K=2$), respectively. The states are labeled by the value of $\delta_\chi/\pi$. Each orange ellipse represents a singlet bond formed by two $S$=1/2 spins or its equivalent in the case of $S$=1. The blue arrow represents the conduction electron spin, while the short (long) red arrow represents $S$=1/2 ($S$=1) impurity spin. (d) Schematic pictures illustrating the antiferromagnetic (left) and ferromagnetic (right) spin current correlations. The symbol $\otimes$ and $\odot$ denote two opposite spin polarizations of the spin current.

  • Figure 2

    (Color online) Holon phase shift and Kondo correlations. (a) Evolution of the holon energy levels with $T_{\rm~K}/J_H$ for $R=1$, $2$ and $1.68$. (b)-(d) The holon phase shift, spin current correlation, and local Kondo correlation as functions of $T_{\rm~K}/J_H$. Data for $R=1$, $2$ and $1.68$ are shown with red, green and blue points, respectively.

  • Figure 3

    (Color online) Spin correlations and spinon density of states. (a) The spinon hopping amplitude as a function of $T_{\rm~K}/J_H$ for $R=1$, $2$ and $1.68$. (b) The spin correlation function $\left\langle~\boldsymbol{S}_i\cdot~\boldsymbol{S}_j\right\rangle$ as a function of $T_{\rm~K}/J_H$ for $R=1$, $2$ and $1.68$. The inset shows its derivative with respect to $T_{\rm~K}/J_H$ for $R=1$ and $2$. (c), (d) Intensity plots of the spinon's density of states, $-\sum_h~G_b''(h,\omega)/(3\pi)$, for $R=1.68$ and $R=2$, showing their evolutions with $T_{\rm~K}/J_H$.

  • Figure 4

    (Color online) Renormalization group flow. (a) The RG flow of the coupling constant $g_{\rm~K}'$ for different $R$, assuming that the local Kondo coupling $g_{\rm~K}$ flows to a fixed point $g^*=0.5$. The initial value is $g_{\rm~K}=0.2$, corresponding to $J_{\rm~K}=0.4$ in the large-$N$ calculations. (b) $g_{\rm~K}'$ as a function of $R$ at a fixed scale $l=10$ with $g^*=0.5$. (c) Intensity plot of $\text{sgn}(g_{\rm~K}')\ln~(|g_{\rm~K}'|+1)$ on the $R$-$g^*$ plane at the same scale $l=10$, showing the sign change of $g_{\rm~K}'$ with varying $R$.

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