Luttinger liquids with curvature: Density correlations and Coulomb drag effect

We consider the effect of the curvature in fermionic dispersion on the observable properties of Luttinger liquid (LL). We use the bosonization technique where the curvature is irrelevant perturbation, describing the decay of LL bosons (plasmon modes). When possible, we establish the correspondence between the bosonization and the fermionic approach. We analyze modifications in density correlation functions due to curvature at finite temperatures, $T$. The most important application of our approach is the analysis of the Coulomb drag by small momentum transfer between two LLs, which is only possible due to curvature. Analyzing the ac transconductivity in the one-dimensional drag setup, we confirm the results by Pustilnik et al. for $T$ dependence of drag resistivity, $R_{12}\sim T^2$ at high and $R_{12}\sim T^5$ at low temperatures. The bosonization allows for treating both intra- and interwire electron-electron interactions in all orders, and we calculate exact prefactors in the low-$T$ drag regime. The crossover temperature between the two regimes is $T_1\sim E_F\Delta$, with $\Delta$ relative difference in plasmon velocities. We show that $\Delta \ne 0$ even for identical wires, due to lifting of degeneracy by interwire interaction, $U_{12}$, leading to crossover from $R_{12}\sim U_{12}^2{T}^2$ to $R_{12}\sim T^5∕U_{12}$ at $T\sim U_{12}$.

Figure 1

Feynman diagrams representing nonlinear response of the fermionic system. Two three-tail fermion loops correspond to a boson decay process, see the text for discussion.

Figure 2

First Coulomb drag diagrams in the fermionic formalism and their bosonization counterpart, see the text for discussion.

Figure 3

Two boson diagrams for ac conductivity, obtained with the use of continuity relation and equivalent to Fig. 4.

Figure 4

Two boson loop diagrams formally contributing to ac conductivity. The labels $\theta \varphi$, etc., at boson lines refer to the boson fields as discussed in text.

Figure 5

Skeleton diagrams depicting the bosonic processes contributing to drag effect at lowest temperatures.

Figure 6

RPA processes, which do not contribute to drag conductivity.

Figure 7

Four cross sections corresponding to two above skeleton graphs.

Figure 8

The leading contribution to the low-$T$ drag coefficient is given by diagrams with three propagators in a cross section.

Figure 9

First order in curvature contribution to the vertex part in the above diagram.

Figure 10

Singular diagrams.