Kondo effect in a quantum wire with spin-orbit coupling

The influence of spin-orbit interactions on the Kondo effect has been under debate recently. Studies conducted recently on a system composed of an Anderson impurity on a two-dimensional electron gas with a Rashba spin orbit have shown that it can enhance or suppress the Kondo temperature $\left(T_{\mathrm{K}}\right)$, depending on the relative energy level position of the impurity with respect to the particle-hole symmetric point. Here, we investigate a system composed of a single Anderson impurity, side coupled to a quantum wire with spin-orbit coupling (SOC). We derive an effective Hamiltonian in which the Kondo coupling is modified by the SOC. In addition, the Hamiltonian contains two other scattering terms, the so-called Dzyaloshinskii-Moriya interaction, known to appear in these systems, and another one describing processes similar to the Elliott-Yafet scattering mechanisms. By performing a renormalization group analysis on the effective Hamiltonian, we find that the correction on the Kondo coupling due to the SOC favors the enhancement of the Kondo temperature even in the particle-hole symmetric point of the Anderson model, agreeing with the numerical renormalization group results. Moreover, away from the particle-hole symmetric point, $T_{\mathrm{K}}$ always increases with the SOC, accordingly with a previous renormalization group analysis.

Figure 1

Schematic representation of a quantum dot, side coupled to a quantum wire with spin-orbit interaction. The wire is assumed to lie along the $x$ direction. $V_k$ represents the hopping of electrons from the quantum into the wire.

Figure 2

(a) Spin-orbit bands for the conduction electrons. At low temperature, the allowed processes are those involving energies close to the Fermi level ${\varepsilon }_F$ (horizontal dashed line). The magenta and purple arrows exemplify, respectively, the intraband (forward) and intraband (backward) scatterings. (b) and (c) are representative scattering diagrams describing typical processes contained in the Hamiltonians (28) and (30), respectively.

Figure 3

(a) Scaled Kondo temperature vs ${\gamma }_F/U$ for different values of ${\varepsilon }_d$ and $U=0.1$. ${\varepsilon }_d=-0.5U$ corresponds exactly to the particle-hole symmetric point of the Anderson model. Note the different behavior of $T_{\mathrm{K}}$ for ${\varepsilon }_d$ above and below 0.05. $T_{\mathrm{K}}^0$ is the Kondo temperature calculated in the absence of the SO interaction, $\gamma =0$. (b) $log\left(T_{\mathrm{K}}/T_{\mathrm{K}}^0\right)$ vs ${\gamma }_F/U$ (symbols). The solid lines show linear functions connecting the first and the last points of each data set, serving as a guide to the eyes. These lines suggest that $T_{\text{K}}$ depends on ${\gamma }_F$ exponentially as $T_{\mathrm{K}}=T_{\mathrm{K}}^{0}exp\left(a{\gamma }_F^2\right)$, in which $a$ is a function of ${\varepsilon }_d$. (c) $a/U^2$ vs ${\varepsilon }_d/U$ extracted from the results of (b).