Spin-charge mixing effects on resonant tunneling in a polarized Luttinger liquid

We investigate spin-charge mixing effect on resonant tunneling in spin-polarized Tomonaga-Luttinger liquid with double impurities. The mixing arises from Fermi velocity difference between two spin species due to Zeeman effect. Zero bias conductance is calculated as a function of gate voltage $V_g$, gate magnetic field $B_g$, temperature, and magnetic field applied to the system. Mixing effect is shown to cause rotation of the lattice pattern of the conductance peaks in $(V_g,B_g)$ plane, which can be observed in experiments. At low temperatures, the contour shapes are classified into three types, reflecting the fact that effective barrier potential is renormalized towards “perfect reflection,” “perfect transmission,” and magnetic field induced “spin-filtering,” respectively.

Figure 1

(Color online) Schematic figure of spin polarized Tomonaga-Luttinger liquid with two impurities under magnetic field.

Figure 2

Second order conductance correction $G_{\rho }^{\left(2\right)}$ is plotted as a function of gate voltage and magnetic field normalized by ${\delta }_i=\frac{v_i}{K_i}\frac{\pi }{2d}$ for Hubbard model. $K_{\rho }=v_F∕v_{\rho }$, $K_s=v_F∕v_s=1$; (a) $K_{\rho }=1$, $\Delta ∕v_F=0$, (b) $K_{\rho }=0.5$, $\Delta ∕v_F=0$, (c) $K_{\rho }=1$, $\Delta ∕v_F=0.5$, (d) $K_{\rho }=0.5$, $\Delta ∕v_F=0.5$. The temperature is fixed at $T∕\Lambda =0.1$ in all figures. Solid (broken) arrow indicates the direction of peak line of the conductance deviation due to up- (down-) spin resonant tunneling.

Figure 3

$G_{\rho }^{\left(2\right)}$ is plotted as a function of gate voltage and magnetic field normalized by ${\delta }_i=\frac{v_i}{K_i}\frac{\pi }{2d}$ for spin anisotropic interaction parameters. $K_{\rho }=v_F∕v_{\rho }=0.6$, $K_s=v_F∕v_s=1.4$, $\Delta ∕v_F=0.5$; (a) $T∕\Lambda ={10}^{-2}$, (b) $T∕\Lambda ={10}^{-6}$.

Figure 4

Zero bias conductance $G_{\rho }$ is plotted as a function of gate voltage and magnetic field normalized by ${\delta }_i=\frac{v_i}{K_i}\frac{\pi }{2d}$, with $T={10}^{-1}\frac{\pi v_F}{4d}={10}^{-3}\Lambda$ and $(t_{\sigma }^j∕v_{F,\sigma }{)}^2∕\pi ={10}^{-2}$. Different values of interaction and magnetic field are taken for each figure; (a) $K_{\rho }=K_s=1$, $\Delta ∕v_F=0$, (b) $K_{\rho }=K_s=1$, $\Delta ∕v_F=0.3$, (c) $K_{\rho }=0.5$, $K_s=1$, $\Delta ∕v_F=0$, and (d) $K_{\rho }=0.5$, $K_s=1$, $\Delta ∕v_F=0.3$.