Tunneling into and between helical edge states: Fermionic approach

We study the four-terminal junction of spinless Luttinger liquid wires, which describes either a corner junction of two helical edge states of topological insulators or the tunneling from the spinful wire into the helical edge state. We use the fermionic representation and the scattering state formalism, in order to compute the renormalization group (RG) equations for the linear response conductances. We establish our approach by considering a junction between two possibly nonequivalent helical edge states and find an agreement with the earlier analysis of this situation. Tunneling from the tip of the spinful wire to the edge state is further analyzed which requires some modification of our formalism. In the latter case we demonstrate (i) the existence of both fixed lines and conventional fixed points of RG equations, and (ii) certain proportionality relations holding for conductances during renormalization. The scaling exponents and phase portraits are obtained in all cases.

Figure 1

(a) Schematically shown X junction, (b) X junction of the helical edge states, with spin projections explicitly indicated; see also Table 1.

Figure 2

Full scaling curves for conductances $G_R=I_1^{\mathrm{new}}/V_1^{\mathrm{new}}$ and $G_D=I_2^{\mathrm{new}}/V_2^{\mathrm{new}}$, defined in Eq. (4), and the Luttinger parameter $K=0.4$. Slight difference in the initial conductances results in the opposite qualitative behavior. Panels (a) and (b) show RG flows tending to the FP of the perfect transmission in the vertical and horizontal directions, respectively, in terms of Fig. 1. The nonmonotonic behavior stems from the RG flow passing near the FP of the saddle-point type at $a=b=0$ in Fig. 3.

Figure 3

RG flows (red lines, the direction of flow shown by arrows) and fixed points are shown for different values of interaction in the edges states. Panel (a) corresponds to Luttinger parameters $K_1=0.3,K_2=0.5$ and panel (b) is for $K_1=1.2,K_2=0.5$. The disappearance of three nonuniversal unstable FPs is visible in panel (b), which is accompanied by the change of the character of the lower red FPs.

Figure 4

Phase portrait for conductances in a setup with different values of interaction in the edges states, characterized by Luttinger parameters, $K_{1,2}$. Depending on the sign of $(1-K_1)(1-K_2)$ one has either one or three stable fixed points, as described in text.

Figure 5

RG phase portrait for tunneling between the spinful wire and helical edge state. The brown region at $K_2<1$ is the stability region for the FP corresponding to the absence of tunneling. In the blue region the situation is characterized by one stable FP and a stable fixed line, as shown in Fig. 6. In the white region only the fixed line $b=-1$ is stable.

Figure 6

Two examples of RG flows are shown. Typical behavior in the brown region of Fig. 5 is shown in panel (a) for $K_1=1.55,K_2=0.51$. The situation in the blue region of Fig. 5 is qualitatively reproduced in panel (b) at $K_1=1.38,K_2=1.03$.