Impurity scattering in a Luttinger liquid with electron-phonon coupling

We study the influence of electron-phonon coupling on electron transport through a Luttinger liquid with an embedded weak scatterer or weak link. We derive the renormalization group (RG) equations, which indicate that the directions of RG flows can change upon varying either the relative strength of the electron-electron and electron-phonon coupling or the ratio of Fermi to sound velocities. This results in a rich phase diagram with up to three fixed points: an unstable one with a finite value of conductance and two stable ones, corresponding to an ideal metal or insulator.

Figure 1

Weak tunneling, ${\gamma }_{+}$, and weak scattering, ${\gamma }_{-}$, exponents (solid lines) as functions of the Luttinger parameter $K<1$ for a weak, ${\alpha }_{\mathrm{ph}}<1$, in (a), or strong, ${\alpha }_{\mathrm{ph}}>1$, in (b), el-ph coupling (we used ${\alpha }_{\mathrm{ph}}=0.7$ and ${\alpha }_{\mathrm{ph}}=1.5$). Both ${\gamma }_{\pm }$ change sign for any ${\beta }_{{}_{\mathrm{F}}}$ for ${\alpha }_{\mathrm{ph}}<1$ and for ${\beta }_{{}_{\mathrm{F}}}<{\beta }_{{}_{\mathrm{F}}}^{*}$, Eq. (14), for ${\alpha }_{\mathrm{ph}}>1$ (we used ${\beta }_{{}_{\mathrm{F}}}=0.5$ in both cases)—in contrast to the Kane-Fisher exponents ${\mathrm{\gamma ̃}}_{\pm }$ for the phononless LL (dotted lines). A typical behavior of both exponents for ${\alpha }_{\mathrm{ph}}>1$ and ${\beta }_{{}_{\mathrm{F}}}>{\beta }_{{}_{\mathrm{F}}}^{*}$ is shown by dashed lines in (b). In this case the exponent ${\gamma }_{+}$ remains positive within the stability region $K<{\alpha }_{\mathrm{ph}}^{-1/2}$.

Figure 2

Phase diagram: The light-gray area represents the insulating regime. In the dark-gray area there exists a line of unstable fixed points separating the metallic state for a weak scatterer from the insulating phase for a strong one. The hatched area (reachable only for ${\alpha }_{\mathrm{ph}}<{\alpha }_{\mathrm{ph}}^{*}$) corresponds to the purely metallic phase. We used ${\beta }_{{}_{\mathrm{F}}}=1.1$ here.