In several types of materials electrons are confined to one spatial dimension. Examples are highly anisotropic three dimensional crystals, quantum wires based on semiconductor heterostructures or carbon nanotubes, and atomic chains on surfaces. In one dimension the electron-electron interaction of the conduction electrons is of particular importance. The elementary excitations are no longer given by fermionic quasi-particles but are of collective nature. This leads to a variety of observable differences to the behavior of three-dimensional interacting electron systems (so-called Fermi liquids), which are summarized in the (Tomonaga-) Luttinger liquid phenomenology. Of particular interest are the spectral properties of Luttinger liquids which we extensively studied in the past. Currently the focus moved towards transport properties of inhomogeneous Luttinger liquids, e.g. systems with a few impurities, junctions of one-dimensional wires, and chains with contacts to leads, which are also intensively investigated experimentally. Such systems might serve as basic units of an emerging nano-electronics. The methods used are the so-called bosonization, the functional renormalization group, and numerical approaches.
Fixed point structure and renormalization group flow of a Y-junction of three interacting quantum wires with enclosed magnetic flux.