Charge-spin duality in nonequilibrium transport of helical liquids

Nonequilibrium transport properties of the charge and spin sectors of two edges of a quantum spin Hall insulator are investigated theoretically in a four-terminal configuration. A simple duality relation between charge and spin sector is found for two helical Tomonaga Luttinger liquids (hTLLs) connected to noninteracting electron reservoirs. If the hTLLs on opposite edges are coupled locally or nonlocally, the mixing between them yields interesting physics where spin information can be easily detected by a charge measurement and vice versa. Particularly, we show how a pure spin density in the absence of charge current can be generated in a setup that contains two hTLL and one spinful Tomonaga Luttinger liquid in between.

Figure 1

(a) Schematic of the four-terminal setup. At each edge, there is a conducting channel of a hTLL (blue full lines correspond to spin up; red dashed lines to spin down). The two hTLLs are mixed in the junction region and different types of junctions are analyzed; (b) the short junction with two possible single particle scattering terms: (i) spin conserved scattering $t_c$ and (ii) spin-flip scattering $t_s$ and (c) the long junction modeled by a sTLL.

Figure 2

(a) (I) and (III) show band dispersions and chemical potentials of the upper edge and the lower edge, respectively. (II) illustrates the voltage configuration ${\mu }_{1,+}={\mu }_{1,-}={\mathit{eV}}_0/2$ and ${\mu }_{2,+}={\mu }_{2,-}=-{\mathit{eV}}_0/2$, yielding $V_c=V_0$ and $V_s=0$. (b) Similar to (a) with a different voltage configuration ${\mu }_{1,+}={\mu }_{2,-}={\mathit{eV}}_0/2$ and ${\mu }_{2,+}={\mu }_{1,-}=-{\mathit{eV}}_0/2$, giving $V_c=0$ and $V_s=V_0$.

Figure 3

(a) Backscattering (BS) current as a function of charge bias $V_c$ for the voltage configuration of Fig. 2a. The blue dashed line corresponds to the charge current $\langle j_{2,c}^{(0)}\rangle$, while the red solid line additionally includes the finite length corrections. The BS current is plotted in units of $\frac{2{\mathit{et}}_c^2[{\tau }_{\mathrm{cu}}^{(L)}]{}^{g_0+\frac{1}{g_0}}}{\hslash {}^2{\omega }_L}$. (b) BS spin densities $\langle {\rho }_{2,s}^{(0)}\rangle$ (blue dashed line) and $\langle {\rho }_{2,s}\rangle$ (red solid line) as a function of the spin bias $V_s$ for the voltage configuration of Fig. 2b. The BS spin density is plotted in units of $\frac{2t_s^2[{\tau }_{\mathrm{cu}}^{(L)}]{}^{g_0+\frac{1}{g_0}}}{\hslash {}^2{v}_f{\omega }_L}$. (c) Possible two-particle backscattering terms: spin conserved backscattering $g_{1\perp }$ and spin-flip backscattering $g_{\mathit{sf}}$. (d) Voltage dependence of $\langle {\rho }_{2,s}\rangle$ due to the two-particle spin-flip term $g_{\mathit{sf}}$. Here, the BS spin density is plotted in units of $\frac{g_{\mathit{sf}}^{2}d{g}_s{g}_0[{\tau }_{\mathrm{cu}}^{(L)}]{}^{\frac{4}{g_s{g}_0}}}{2(\hslash {\omega }_d^s){}^2}$. In all expressions above, ${\omega }_L=\frac{v_f}{g_{0}L}$, ${\omega }_d^s=\frac{v_f}{g_{s}d}$, and, in the plots, we choose $g_0=0.7$ and $g_s=1$.