Electrons in a one-dimensional system form a quantum liquid which can be described as a Luttinger liquid rather than by Landau's Fermi-liquid theory. The model of a one-dimensional electron gas was developed by Shin'ichiro Tomonaga and Joaquin Luttinger in the early sixties and advanced by many theoreticians since. Due to the large effects of electronic interaction, this new state of matter has radically different properties, such as [1,2,3]

While for free electrons and all other Fermi-liquids the occupation function n(k) exhibits a discontinuity at the Fermi wavevector kF, the Luttinger liquid is described by some power law:


The exponent αis given by the Luttinger liquid parameter Kρ
which characterizes the interaction (Kρ = 1 means no interaction). The single-particle density of states near kF is expected to follow a power-law
which leads to a power-law behavior of the electron tunneling. The transport properties of a Luttinger liquid along the chain direction depend on the scattering mechanism and the band filling 1/n:
describes the temperature-dependent conductivity in the dc limit and
the optical conductivity in the limit of low temperatures. Thus the dc and frequency-dependent transport characteristics of a Luttinger liquid significantly deviate from those of a Fermi liquid. Furthermore, drastic effects of a single impurity on the transport properties of a Luttinger liquid are theoretically expected; in particular, in the case of repulsive interactions the conductance is predicted to vanish as a power of the temperature [4].

It is one of the peculiarities of Luttinger liquids that the elementary excitations of are not quasi-particles with charge e and spin 1/2 but collective charge and spin density fluctuations with bosonic character, so-called spinons and holons. These spin and charge excitations propagate with different velocities which leads to the separation of spin and charge. Figure 1 shows a simple picture of spin-charge separation for a half-filled antiferromagnetic insulator, as it can be observed in a photoemission experiment.


Spin-charge separation for a half-filled one-dimensional antiferromagnetic chain. In the photoemission process an electron is excited and thus a hole (holon) is produced in the chain. Hopping of the hole to the neighboring site or - equally - hopping of the neighboring electron onto the empty site leads to a magnetic disorder (spinon). Further hopping, however, does not cause further disorder in the spin chain. Thus, the photo hole splits into two separate "defects" in the chain which can be observed as two separate particles.

The above characteristics apply for a perfect one-dimensional interacting electron system, e.g., a single one-dimensional metallic system without gaps in the spin and charge sector. If one considers an array of parallel one-dimensional metallic chains, one can increase the transverse interaction between the single chains by changing the distance, and thus one is able to tune the dimensionality. The Luttinger liquid behavior will diminish and some continuous crossover to a Fermi-liquid is expected. The actual behavior, however, has not been systematically explored yet, neither theoretically nor experimentally. Only recently, first attempts were made to describe the coupling between one-dimensional chains [5,6]. The most important term in this regard is the interchain hopping t, but also direct exchange is possible, like in the case of density-density or spin-spin exchange.

In the absence of electronic interactions, the Fermi surface consists of two parallel planes in the case of a one-dimensional system, which are slightly warped as t becomes important. If one is at an energy scale larger than the warping (high temperatures T or large frequencies ω), then the warping is washed out and the system is considered as one-dimensional. The crossover energy is given by

At low energies two- or three-dimensional properties are expected. Interaction between the electrons leads to a renormalization and the crossover takes place at some lower energy. With other words, electronic correlations tend to make the system more one-dimensional since in a Luttinger liquid single-particle excitations are strongly suppressed. In addition, pair hopping and spin-spin and density-density couplings occur. These two-particle processes are of the order of t2 and in general much smaller than the single-particle hopping. The situation is different, however, when the chains are in a Mott insulating state due to commensuration, because single-particle hopping becomes irrelevant since a pair has to be broken. On the other hand, increasing the interaction breaks the one-dimensional Mott gap Δ and drives the system to a gapless higher-dimensional state, i.e., to a metal. Thus the simple dimensional crossover is now replaced by a quantum phase transition for a critical value of the interchain hopping, where the system goes from a one-dimensional Mott insulator to a two-dimensional metal; this is known as deconfinement transition and depicted in Figure.


Schematic representation of the deconfinement. The quantum phase transition takes place at T = 0 as a function of the interchain hopping t⊥. At finite temperatures the presence of this quantum critical point leads to various crossovers between a Mott insulator, Luttinger liquid and two-dimensional metal. Whether such a phase is a Fermi liquid and what are its properties, is one of the main questions.

It is difficult to extract the physical properties in the deconfined phase. The transverse conductivity was computed in the high-temperature high-frequency regime (T>> Δ):

and for the frequency dependence:
The above discussed theoretical predictions for the thermodynamic and spectroscopic properties of a Luttinger liquid can be verified experimentally by various techniques. One of the most promising experiment is photoemission spectroscopy which directly probes the single-particle DOS D(ω) and, in the angle-resolved mode, the single-particle spectral function A(k,ω) [7,8]. Another important technique is the tunneling transport, which enables the detailed study of the low-energy behavior down to the μeV-range [9]. Temperature- and frequency-dependent conductivity measurements also serve as an important probe of the Luttinger liquid properties, as well as NMR and thermal conductivity measurements.

There are different approaches how one-dimensional electron systems can be realized, and the resulting systems can be divided into natural and artificial ones. The first real systems for which one-dimensional physics was evidenced were strongly anisotropic organic and inorganic conductors consisting of atomic chains or ladders. Among the most prominent ones are the organic salts TTF-TCNQ, the (DCNQI)2Cu salts and the "Bechgaard salts" (TMTSF)2X, and (TMTTF)2X [10,11]. Many studies were also carried out on anisotropic inorganic materials [7,8], e.g., the charge-density-wave compounds K0.3MoO3 (the so-called blue bronze), (TaSe4)2I, BaVS3, NbSe3, the one-dimensional metal Li0.9Mo6O17 (purple bronze), alpha′- NaV2O5 and SrNbO3.4, and the spin chain compound SrCuO2, Sr2CuO3. For these materials evidence for strong deviations from conventional 3D physics comes from photoemission [7,8], where the absence of quasiparticle peaks and a suppression of spectral weight at the Fermi energy were found; there exist hints of spinon and holon excitations as well [12], but an unambigous proof of spin-charge separation by photoemission is lacking so far. One of the reasons is that these compounds tend to undergo instabilities like Peierls or Mott-Hubbard transitions which lead to a metal-insulator transition.Indications for spin-charge separation were also concluded from comparison of magnetic with electronic properties and thermal with electronic conduction [11,13,14,15]. Among the most convincing evidences for Luttinger liquid behavior in natural materials is the power-law behavior observed in the optical conductivity of various quasi-one-dimensional organic conductors [16,17].

Recently, the Luttinger liquid picture was also suggested for the description of high-temperature superconductors in their stripe phase [18], where between antiferromagnetic stripes metallic charge stripes are formed, which electronically behave as one-dimensional systems. Within this picture, high-temperature superconductors are thus described as quasi-one-dimensional superconductors, which behave as Luttinger liquids at sufficiently high temperature.

One of the main difficulties in finding unambiguous evidence for Luttinger liquid characteristics in natural one-dimensional metals (which are in fact quasi-one-dimensional since a certain coupling between the one-dimensional entities is always present) is the fact that they tend to undergo phase transitions, like charge-density-wave or spin-density-wave transitions, which mask the Luttinger-liquid signatures one wants to probe. To some extent, this complication can be avoided by the study of artificial one-dimensional electron systems, like fractional quantum Hall effect edge (FQHE) states [9], nanotubes, and quantum wires [19]. Probably the most intriguing approach is the arrangement of metal atoms on surfaces, in particular gold wires on silicon [19].

Contact: M. Dressel