Critical behavior of a point contact in a quantum spin Hall insulator

Abstract

We study a quantum point contact in a quantum spin Hall insulator. It has recently been shown that the Luttinger liquid theory of such a structure maps to the theory of a weak link in a Luttinger liquid with spin with Luttinger liquid parameters ${g}_{\ensuremath{\rho}}=1/{g}_{\ensuremath{\sigma}}=g<1$. We show that for weak interactions, $1/2<g<1$, the pinch-off of the point contact as a function of gate voltage is controlled by a novel quantum critical point, which is a realization of a nontrivial intermediate fixed point found previously in the Luttinger liquid model with spin. We predict that the dependence of the conductance on gate voltage near the pinch-off transition for different temperatures collapses onto a universal curve described by a crossover scaling function associated with that fixed point. We compute the conductance and critical exponents of the critical point as well as the universal scaling function in solvable limits, which include $g=1\ensuremath{-}ϵ$, $g=1/2+ϵ$, and $g=1/\sqrt{3}$. These results, along with a general scaling analysis, provide an overall picture of the critical behavior as a function of $g$. In addition, we analyze the structure of the four-terminal conductance of the point contact in the weak tunneling and weak backscattering limits. We find that different components of the conductance can have different temperature dependences. In particular, we identify a skew conductance ${G}_{XY}$, which we predict vanishes as ${T}^{\ensuremath{\gamma}}$ with $\ensuremath{\gamma}\ensuremath{\ge}2$. This behavior is a direct consequence of the unique edge state structure of the quantum spin Hall insulator. Finally, we show that for strong interactions, $g<1/2$, the presence of spin nonconserving spin-orbit interactions leads to a novel time-reversal-symmetry breaking insulating phase. In this phase, the transport is carried by spinless chargons and chargeless spinons. These lead to nontrivial correlations in the low frequency shot noise. Implications for experiments on HgCdTe quantum well structures will be discussed.