Kondo Effect in the Helical Edge Liquid of the Quantum Spin Hall State

Following the recent observation of the quantum spin Hall (QSH) effect in HgTe quantum wells, an important issue is to understand the effect of impurities on transport in the QSH regime. Using linear response and renormalization group methods, we calculate the edge conductance of a QSH insulator as a function of temperature in the presence of a magnetic impurity. At high temperatures, Kondo and/or two-particle scattering give rise to a logarithmic temperature dependence. At low temperatures, for weak Coulomb interactions in the edge liquid, the conductance is restored to unitarity with unusual power laws characteristic of a “local helical liquid,” while for strong interactions, transport proceeds by weak tunneling through the impurity where only half an electron charge is transferred in each tunneling event.

Figure 1

Temperature dependence of the conductance: the behavior is logarithmic at high temperature $T\gg T^{*}$ and power law at low temperature $T\ll T^{*}$. At $T=0$, a metal-insulator quantum phase transition is driven by Coulomb interactions in the helical liquid: the system is a “Kondo metal” for $K>1/4$ and a “Luttinger liquid insulator” for $K<1/4$. The Fano factor $e^{*}$ is defined as the ratio between shot noise and current and reflects the charge of the current-carrying excitations.

Figure 2

Strong coupling regime: the Kondo singlet effectively removes one site from the system. (a) Luttinger liquid with $K_{\rho }<1$: a punctured 1D lattice is disconnected, (b) QSH edge liquid with $K>1/4$: the edge liquid follows the boundary of the deformed 2D lattice. (c) Half-charge tunneling for $K<1/4$ by flips of the Ising order parameter $m$.