Tunneling into a Luttinger Liquid Revisited

We study how electron-electron interactions renormalize tunneling into a Luttinger liquid beyond the lowest order of perturbation in the tunneling amplitude. We find that the conventional fixed point has a finite basin of attraction only in the point contact model, but a finite size of the contact makes it generically unstable to the tunneling-induced breakup of the liquid into two independent parts. In the course of renormalization to the nonperturbative-in-tunneling fixed point, the tunneling conductance may show a nonmonotonic behavior with temperature or bias voltage.

Figure 1

Junction between a quantum wire (horizontal) and a tunnel electrode (vertical). The arrows denote the scattering processes (and their amplitudes) for incoming electrons.

Figure 2

Renormalization group flows (schematic) for the point tunnel junction. Two stable FPs at $\rho =0$ and $\rho =1$ describe the homogeneous wire and the breakup of the wire into two independent semi-infinite pieces, respectively. Tunneling into the wire is blocked at both points.

Figure 3

Lowest-order scattering processes leading to the renormalization of the reflection amplitude $r$. Dots: tunneling amplitude $t_0$. Wavy lines: bare interaction $\alpha$ between the right and left movers. The straight single and double lines denote the electron Green function in the wire and in the tunnel electrode, respectively.

Figure 4

Skeleton diagrams (a) and (b) contribute to the exact-in-$\rho$ $\beta$ function at second order in $\alpha$, diagrams (c)–(e) do not. The thick straight lines denote the noninteracting electron Green function fully dressed by the tunneling vertices.

Figure 5

Renormalization group flows for the tunnel junction in the $G_t-G_w$ plane for $\alpha =0.15$. The only stable FP is at $G_t=G_w=0$ (breakup of the junction into three parts). The unstable FP at $G_t=G_t^c$ corresponds to the point ${\rho }_c$ in Fig. 2. The dashed line separates the regions in which $G_t$ grows (above) and falls off (below) with decreasing energy.

Figure 6

Left panel: schematics of the renormalization of the tunnel conductance $G_t$ on the log-log scale for the generic tunnel contact with $\phi \ne 0$ (solid lines) and the point contact (dashed). Curves I and IV (II and III) correspond to the bare conductance $G_{t0}$ below (above) the dashed line in Fig. 5. The arrow indicates the point at which the contact is reflectionless. The flow of $G_t$ stops at the largest scale of $ϵ$ among $T$, $v/L$, and the voltage between the tunnel electrode and the wire. Right panel: the tunnel conductance of the point contact for $\alpha =0.4$ and $G_{t0}$ above (upper curve) and just below (lower) the critical conductance $G_t^c\simeq 0.45$ (dashed line). The criticality retards the development of the ZBA.