ScienceWeek

CONDENSED MATTER: ON TOMONAGA-LUTTINGER LIQUIDS

The following points are made by Marc Bockrath (Nature 2003 426:511):

1) In physical systems, scale is usually of consequence. Materials of greatly reduced dimension often behave in qualitatively different ways from their macroscopic counterparts. Much of this new behavior arises in systems so small that the conduction electrons are no longer free to move in all three dimensions but are effectively confined to lower-dimensional spaces. In these low-dimensional systems, the effects of interactions between electrons can lead to fascinating new phenomena not observed in higher dimensions. For example, striking results are reported by Ishii et al(2) from photoemission studies of carbon nanotubes, which are effectively one-dimensional systems. The data show a marked departure from the behavior expected for non-interacting electrons, and provide a spectacular demonstration of the effects of interactions in low-dimensional systems.

2) Theoretical calculations indicate that in a 1-dimensional electron system such as a single-walled carbon nanotube (SWNT), the electrons exist in a state called a "Tomonaga-Luttinger liquid" (TLL) (3). In a TLL, the low-energy excitations of its electrons are collective and sound-like, involving the correlated motion of many electrons, rather than the single-electron-like excitations found in conventional three-dimensional metals. This has profound effects on many properties of the system. One such property is the "single-particle spectral function", a quantity that is used to predict the results of experiments when an electron is suddenly added to or removed from an electron system. For a three-dimensional metal, the spectral function is expected to be nearly constant near the Fermi level (the surface of the electron sea, at zero temperature). In contrast, in a TLL, this addition or removal becomes difficult for electrons near the Fermi level, because it requires a complex rearrangement of all the other electrons in the system. As a result, the spectral function approaches zero at the Fermi level, following a power law with an exponent that depends on the strength of the interactions in the TLL. These interactions are conventionally parametrized by a dimensionless constant g, ranging between zero and one for repulsive interactions.

3) Numerous experiments to probe this behavior have been performed on a variety of one-dimensional systems, from quantum Hall edge states(4) to organic conductors(5). SWNTs in particular have proved to be an excellent one-dimensional system, because electrons are able to travel large distances inside them without scattering. By attaching leads to the tubes, electrons can be injected into them, to probe the nanotube's single-particle spectral function. In such experiments, TLL behavior should appear as a power-law dependence of the nanotube conductance on the applied voltage or temperature. Measurements on small bundles and single nanotubes have been in relatively good agreement with these predictions. However, transport measurements necessarily require at least two contacts, and electrons must travel along the nanotubes, potentially complicating the interpretation of the data.

References (abridged):

1. Anderson, P. W. Science 177, 393-396 (1972)

2. Ishii, H. et al. Nature 426, 540-544 (2003)

3. Voit, J. Rep. Prog. Phys. 57, 977-1116 (1995)

4. Chang, A. M., Pfeiffer, L. N. & West, K. W. Phys. Rev. Lett. 77, 2538-2541 (1996)

5. Sekiyama, A., Fujimori, A., Aonuma, S., Sawa, H. & Kato, R. Phys. Rev. B 51, 13899-13902 (1995)

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