Spin-Charge Separation in the Quantum Spin Hall State

The quantum spin Hall state is a topologically nontrivial insulator state protected by the time-reversal symmetry. We show that such a state always leads to spin-charge separation in the presence of a $\pi$ flux. Our result is generally valid for any interacting system. We present a proposal to experimentally observe the phenomenon of spin-charge separation in the recently discovered quantum spin Hall system.

Figure 1

Four different adiabatic processes from ${\varphi }_{\uparrow }={\varphi }_{\downarrow }=0$ to ${\varphi }_{\uparrow }={\varphi }_{\downarrow }=\pm \pi$. The red (blue) curve stands for the flux ${\varphi }_{\uparrow (\downarrow )}(t)$, respectively. The symbol “⊙” (“$\otimes$”) represents increasing (decreasing) fluxes, and the arrows show the current into and out of the Gaussian loop, induced by the changing flux. Charge is pumped in the processes with ${\varphi }_{\uparrow }(t)=-{\varphi }_{\downarrow }(t)$, while spin is pumped in those with ${\varphi }_{\uparrow }(t)={\varphi }_{\downarrow }(t)$.

Figure 2

Local density of states in the core of a flux tube. The dark (blue) color shows the bulk energy gap and the red line shows the midgap states. (a),(b) The evolution of the midgap state upon spin $\Gamma$-flux threadings, with $\Gamma ={\sigma }_z\otimes \mathbb{I}$ and $\Gamma =-({\sigma }_z+{\sigma }_x)\otimes \mathbb{I}/\sqrt{2}$, respectively, in the representation used by Ref. 6. (c) The case of charge flux threading. The two midgap states cross at $\varphi =\pi$. The spatial distribution of the midgap state at $\varphi =\pi$ is shown in (d), in which the zero of $x$ axis corresponds to the position of the flux tube. Here and below, all the calculations are done for the HgTe model of $d=70\AA$ quantum well, with a lattice constant of $a=30\AA$.

Figure 3

(a) The superconductor-quantum well hybrid structure, with the $\pi$ flux tubes generated by a perpendicular magnetic field. (b) The splitting of two midgap states upon increasing $\lambda$. (c) The spatial distribution of the two midgap states for $\lambda =18\mathrm{nm}$. (d) Illustration of the measured charge $Q$ (red solid line) and intensity of ESR signal $I_R=I({\omega }_R)$ at resonance frequency ${\omega }_R=(E_2-E_1)/\hslash$ (blue dashed line) as functions of the chemical potential.