Transport in Quasi One-Dimensional Systems

The interplay of Umklapp scattering from a periodic potential and other scattering processes determine the conductivity of (quasi) one-dimensional metals. We show that the transport at finite temperature is qualitatively and quantitatively strongly influenced by a number of approximate conservation laws. Typically, not the strongest but the second strongest scattering mechanism determines the dcconductivity. We discuss the optical conductivity both of strongly anisotropic, quasi one-dimensional Fermi liquids and of Luttinger liquids. Thanks to the miracles of bosonization, conformal field theory, Bethe ansatz and renormalization group, the theoretical description of one-dimensional systems is one of the best developed areas in the theory of strongly correlated systems [1]. The enormous theoretical advance is contrasted by failure to understand certain experimental quasi one-dimensional systems like the Bechgaard salts even on a qualitative level [2]. In recent years, carbon nanotubes have proven to be an almost ideal one-dimensional system where theoretical concepts can be tested in some detail. Astonishingly, the theory of finite temperature transport in clean onedimensional (1d) systems is not developed very far. The technical reason for this is that in a pure Luttinger-liquid the conductivity is infinite and σ is therefore a singular function of irrelevant operators and naive perturbation theory can be problematic and misleading. In Ref. [3], we argued that main features of the optical conductivity σ(ω, T ) can be understood from the analysis of certain approximate conservation laws and proposed to calculate σ from a hydrodynamic theory of the corresponding slowly decaying modes. The main result is, that in many experimentally relevant situations, the strongest scattering process alone cannot lead to a relaxation of the current due to some conservation law. As a consequence, temperature dependence of the dc-conductivity is often determined by the second strongest scattering processes. This point of view has been tested numerically in [4]. We will first discuss in some detail the role of approximately conserved quantities for a quasi one-dimensional Fermi liquid, afterwards we will shortly review the results for Umklapp scattering in a Luttinger liquid and discuss how the results generalize for other scattering mechanisms.