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

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 $g_{\rho }=1/g_{\sigma }=g<1$. 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 $g=1-ϵ$, $g=1/2+ϵ$, 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 $T^{\gamma }$ with $\gamma \ge 2$. 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.

Figure 1

A quantum point contact in a QSHI, controlled by a gate voltage $V_G$. In (a) $V_G<V_G^{\ast }$ and the point contact is pinched off. The spin filtered edge states are perfectly reflected. In (b) $V_G>V_G^{\ast }$ and the point contact is open. The edge states are perfectly transmitted. In (c) we plot the conductance (later defined as $G_{XX}$) as a function of $V_G$ for different temperatures. As the temperature is lowered, the pinch-off curve sharpens up with a width $T^{\alpha }$. The curves cross at a critical conductance $G^{\ast }$, and the shape of the curve has the universal scaling form (1.1). The plotted curves are based on Eq. (3.9), valid for $g=1-ϵ$, which is computed in Sec. 3C.

Figure 2

Schematic representation of tunneling processes in [(a)–(c)] the CC phase (small $v$) and [(c)–(e)] the II phase (small $t$). (a) and (d) describe single electron processes, while the others are two particle processes. The duality relating $v_e\leftrightarrow t_e$, $v_{\rho }\leftrightarrow t_{\sigma }$, and $v_{\sigma }\leftrightarrow t_{\rho }$ can clearly be seen.

Figure 3

(a) Positions of the minima of $V\left({\theta }_{\rho },{\theta }_{\sigma }\right)$ in Eq. (2.20). When the minima are deep instanton tunneling events between the minima, denoted by $t_e$, $t_{\rho }$, and $t_{\sigma }$, correspond to the transfer of charge and spin across the junction and define the dual theory [Eqs. (2.26, 2.27)]. (b) Positions of the minima of $V\left({\tilde{\theta }}_{\rho },{\tilde{\theta }}_{\sigma }\right)$ in the dual theory [Eqs. (2.26, 2.27)]. Instanton process $v_e$, $v_{\rho }$, and $v_{\sigma }$ correspond to backscattering of charge and spin in the original theory. (c) The IC phase viewed from the CC limit. When $v_{\rho }$ is large and $v_{\sigma }=0$, the minima of $V\left({\theta }_{\rho },{\theta }_{\sigma }\right)$ in Eq. (2.20) are on one-dimensional valleys and define the IC phase. When $v_{\sigma }$ is small but finite the valleys have a periodic potential ${\tilde{v}}_{\sigma }{\tau }^z\text{cos}{\theta }_{\sigma }$, with opposite signs ${\tau }^z=\pm 1$ in neighboring valleys. Instanton tunneling processes between the valleys, denoted ${\tilde{t}}_{\rho }$, switch the sign of ${\tau }^z$. (d) The IC phase viewed from the II limit, in which $t_{\rho }=0$ and $t_{\sigma }$ is large. The valleys have periodic potential ${\tilde{t}}_{\rho }{\tau }^x\text{cos}{\tilde{\theta }}_{\rho }$ with ${\tau }^x=\pm 1$, whose sign is switched by instanton processes ${\tilde{v}}_{\sigma }$.

Figure 4

Tunneling processes in the CC limit allowed by spin nonconserving spin-orbit interactions. $v_{\text{so}}$ is a single particle process where a single spin is flipped, while $v_{\text{sf}}$ is a two particle process, flipping two spins.

Figure 5

Minima of the potential $V\left({\theta }_{\rho },{\phi }_{\sigma }\right)$ in Eq. (2.47). Large $v_{\rho }$ and $v_{\text{so}}$ define the time-reversal breaking insulating phase. Instanton processes ${\tilde{t}}_{\rho }$ and ${\tilde{t}}_{\sigma }$ restore time-reversal invariance. They correspond to tunneling of spinless chargons or chargeless spinons.

Figure 6

(a) Phase diagram for a point contact in a QSHI as a function of the Luttinger liquid parameter $g$. The arrows indicate the stability of the CC, II, CI, and IC phases, as well as the critical fixed point P. This figure assumes spin conservation. In the presence of spin-orbit interactions, the IC phase is unstable for $g<1/2$. This leads to the TBI phase discussed in Sec. 2B5. (b) Conductance $G^{\ast }$ of the critical fixed point as a function of $g$. The curve is a fit, which incorporates the data in Eq. (3.7). (c) Critical exponent ${\alpha }_g$ as a function of $g$. The curve is a fit incorporating the data in Eq. (3.8). $g$ is plotted on a logarithmic scale in all three panels to emphasize the $g\leftrightarrow 1/g$ symmetry.

Figure 7

The universal scaling function ${\mathcal{G}}_g\left(X\right)$ for (a) $g=1-ϵ$ [Eq. (3.9)] and (b) $g=1/2+ϵ$ [Eq. (3.10)]. In (b) the solid line is $ϵ\to 0$, and the dashed line shows the approximate behavior for $ϵ\sim .02$.

Figure 8

(a) Minima of the periodic potential $V\left(\mathbf{r}\right)$ in Eq. (3.13). (b) Minima of $V\left(\mathbf{k}\right)$ in dual theory (3.16). When $g_{\sigma }=3g_{\rho }$ both periodic potentials have triangular symmetry at the critical point, which implies that the mobility ${\mu }_{\alpha \beta }^{\ast }$ is isotropic. This occurs at $g=1/\sqrt{3}$.

Figure 9

Feynman diagrams for the single electron Green’s function. The dashed line is the interaction $u_2\left({\psi }_{\text{in}}^{†}{\psi }_{\text{in}}\right)\left({\psi }_{\text{out}}^{†}{\psi }_{\text{out}}\right)$. The exchange diagram (a) vanishes because it involves $S_{kk}$ and diagrams (b) and (c) cancel one another. (d) and (e) lead to a logarithmic correction to the $S$ matrix given in Eq. (3.38).

Figure 10

Renormalization group flow diagram for the transmission probabilities $\mathcal{T}$, $\mathcal{R}$, and $\mathcal{F}$ based on Eq. (3.40) represented in a ternary plot. The CC, II, and P fixed points of interest in this paper, which have $\mathcal{F}=0$ are on the bottom of the triangle.

Figure 11

Renormalization group flow diagram characterizing the critical fixed point P for $g=1/2+ϵ$. When ${\tilde{v}}_{\sigma }$ and ${\tilde{t}}_{\rho }$ are small, the flows are given by Eq. (3.48). On the axis ${\tilde{v}}_{\sigma }=0$ the fermionization procedure outlined in Sec. 3D2 determines the entire crossover between the IC and CC fixed points. A similar theory describes the crossover between the IC and II fixed points for ${\tilde{t}}_{\rho }=0$.