Dephasing and Weak Localization in Disordered Luttinger Liquid

We study the transport properties of interacting electrons in a disordered quantum wire within the framework of the Luttinger liquid model. The conductivity at finite temperature is nonzero only because of inelastic electron-electron scattering. We demonstrate that the notion of weak localization is applicable to the strongly correlated one-dimensional electron system. We calculate the relevant dephasing rate, which for spinless electrons is governed by the interplay of electron-electron interaction and disorder, thus vanishing in the clean limit.

Figure 1

Diagrams describing the leading WL correction to the conductivity of Luttinger liquid for ${\tau }_{\varphi }^{\mathrm{WL}}\ll \tau$. The dashed lines represent backscattering off impurities. The current vertices are dressed by impurity ladders. The diagrams are understood as fully dressed by $e\mathrm{\text{−}}e$ interactions.

Figure 2

Illustration of electron dynamics governing the WL and dephasing: Time-reversed paths $x_f(t_1)$ (solid lines) and $x_b(t_1)=x_f(t-t_1)$ (dashed lines) on which the interaction-induced action $S$ that yields dephasing of the Cooperon is accumulated. Dotted lines: the propagation of dynamically screened interaction. The interaction may change the direction of propagation upon scattering off disorder (as marked by a cross). Each interaction line gives a contribution to $S$ proportional to $(N_f-N_b{)}^2$, where $N_{f,b}$ is the number of its intersections with the forward ($f$) and backward ($b$) paths. One sees that $N_f\ne N_b$ only due to impurity scattering in the interaction propagator. Interaction and electron lines lying on top of each other do not yield dephasing because of the HF cancellation.

Figure 3

Schematic behavior of $\sigma (T)$ on the log-log scale. Dotted line: the $T$-dependent Drude conductivity [7, 11]. Below $T_1/\alpha$ the WL correction, Eq. (10), dominates $d\ln \sigma /d\ln T$. Below $T_1$ the localization becomes strong.