Tomonaga-Luttinger liquid

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A Tomonaga-Luttinger liquid (named after Shin'ichirō Tomonaga and Joaquin Mazdak Luttinger ), often simply referred to as a Luttinger liquid , is a theoretical model for describing interacting electrons (or other fermions ) in a one-dimensional electrical conductor ( e . B. quantum wires such as carbon nanotubes ). Such a model is necessary because the commonly used model of the Fermi fluid fails in one dimension ( Peierls instability ).

It is believed that the Luttinger model describes the universal behavior at low frequencies (long wavelengths ) of any one-dimensional system of interacting fermions (unless it has made a phase transition to another state).

Tomonaga studied one-dimensional systems of charged fermions in 1950 and predicted their description by bosons (as already, as later proved, Pascual Jordan in the 1930s in an attempt at a neutrino theory of light). Luttinger presented in 1963 (without knowing Tomonaga's work) a special exactly solvable model and Elliott Lieb and Daniel Mattis clarified its exact solvability through bosonization. The name Luttinger Fluid was coined in 1981 by F. Duncan M. Haldane .

properties

The essential properties of a Luttinger liquid include:

The response of the charge or particle density to an external disturbance are density waves ( plasmons ), the speed of which is determined by the strength of the interaction and the mean density. For non-interacting systems, this speed of propagation is equal to the Fermi speed , while it is greater (smaller) than this for repulsive (attractive) interaction between fermions.
There are also spin density waves , the speed of which corresponds to the Fermi speed as a first approximation. These propagate independently of the charge density waves . One speaks therefore of spin-charge separation .
Charge and spin waves are separate elementary excitations of the Luttinger liquid, in contrast to the quasiparticles of the Fermi liquid , which have both spin and charge. The simplest mathematical description is done using these waves. You solve the one-dimensional wave equation , and much of the work is to transform back to get the properties of the particles. Another difficulty is the treatment of defects and other cases where the backscatter (Engl. Backscattering ) plays an essential role.
Even at absolute zero the momentum distribution of the particles does not change abruptly anywhere, in contrast to the Fermi liquid, where its discontinuity defines the Fermi surface .
The momentum-dependent spectral function has no quasiparticle peak (i.e. no vertex whose width above the Fermi level is much smaller than the excitation energy). Instead, there is an algebraic singularity with a non-universal exponent that depends on the strength of the interaction.
In the vicinity of imperfections, the usual Friedel oscillations of the charge density occur with a wave vector of . For large distances from the impurity, these disappear , whereby the exponent depends on the interaction ( for a Fermi liquid). ${\ displaystyle 2k_ {F}}$ ${\ displaystyle x}$ ${\ displaystyle \ sim 1 / | x | ^ {g}}$ ${\ displaystyle g}$ ${\ displaystyle g = 1}$
At low temperatures, the scattering of these Friedel oscillations is so strong that the renormalized effective strength of the defect becomes infinite and thus pinches the quantum wire. More precisely: the conductivity tends to zero with decreasing temperature and applied voltage (and follows a power law, the exponent of which depends on the interaction)
The tunnel rate into the Luttinger fluid is also suppressed at low voltages and temperatures.

Applications

The physical systems that are believed to be described with the Luttinger model include:

artificial quantum wires (one-dimensional electron strips) that are generated, for example, with the help of a gate voltage in a two-dimensional electron gas (or lithographically , or atomic force microscope, etc.)

Electrons in carbon nanotubes

Electrons in edge states of the quantum Hall effect (and its fractional variant)

Electrons (organic molecular crystals specific, for example) along one-dimensional molecular chains hop (engl. Hopping )

fermionic atoms in quasi-one-dimensional atom traps

The proof of the characteristic properties of a Luttinger liquid in these systems is a current research area in experimental solid state physics .

literature

Vieri Mastropietro, Daniel Charles Mattis : Luttinger Model. The First 50 Years and Some New Directions . World Scientific, 2013, ISBN 978-981-4520-71-3 ( abstract at World Scientific ).
Sebastian Mietke: Scanning tunnel microscopy and spectroscopy on Au / Ge (001) nanowires . A model system of the Luttinger fluid. Kassel University Press, 2014, ISBN 978-3-86219-724-8 ( limited preview in Google book search).
HJ Schulz, G. Cuniberti, P. Pieri: Fermi liquids and Luttinger liquids . In: G. Morandi et al. Eds. (Ed.): Field Theories for Low-Dimensional Condensed Matter Systems . Springer, 2000, ISBN 3-540-67177-3 , arxiv : cond-mat / 9807366 .
Johannes Voit: One-dimensional Fermi liquids . In: Rep. Prog. Phys . tape 58 , 1995, pp. 977–1116 , doi : 10.1088 / 0034-4885 / 58/9/002 .

Johannes Voit: A brief introduction to Luttinger liquids . In: Proceedings of the International Winterschool on Electronic Properties of Novel Materials . Kirchberg March 2000, arxiv : cond-mat / 0005114 .

K. Schonhammer: Physics in one dimension: theoretical concepts for quantum many-body systems . In: J. Phys . Condens. Matter 25, 2013, p. 25 , arxiv : 1212.1632 .

Web links

The Net Advance of Physics. Luttinger Liquids. MIT, accessed December 22, 2014 (bibliography).
Nicholas T. Brönn: Luttinger Liquids. (PDF) December 11, 2007, accessed December 22, 2014 (English).
K. Schönhammer: Interacting fermions in one dimension . The Tomonaga-Luttinger model. 1998, arxiv : cond-mat / 9710330 (unpublished preprint).

Individual evidence

↑ Sin-itiro Tomonaga : Remarks on Bloch's Method of Sound Waves applied to Many-Fermion Problems . In: Prog. Theor. Phys . tape 5 , no. 4 , 1950, p. 544-569 , doi : 10.1143 / ptp / 5.4.544 .
^ ^A ^b Daniel C. Mattis , Elliott H. Lieb : Exact Solution of a Many-Fermion System and Its Associated Boson Field . In: J. Math. Phys . tape 6 , no. 2 , 1965, p. 304–312 , doi : 10.1063 / 1.1704281 ( limited preview at SpringerLink [accessed on December 22, 2014]).
↑ JM Luttinger : An exactly soluable model of a many-fermion system . In: J. Math. Phys . tape 4 , no. 9 , 1963, pp. 1154-1162 , doi : 10.1063 / 1.1704046 .
^ FDM Haldane : 'Luttinger liquid theory' of one-dimensional quantum fluids . I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas. In: J. Phys. C . tape 14 , no. 19 , 1981, p. 2585-2919 , doi : 10.1088 / 0022-3719 / 14/19/010 .

[Tomonaga-1] Sin-itiro Tomonaga : Remarks on Bloch's Method of Sound Waves applied to Many-Fermion Problems . In: Prog. Theor. Phys . tape 5 , no. 4 , 1950, p. 544-569 , doi : 10.1143 / ptp / 5.4.544 .

[Lieb-2] A ^b Daniel C. Mattis , Elliott H. Lieb : Exact Solution of a Many-Fermion System and Its Associated Boson Field . In: J. Math. Phys . tape 6 , no. 2 , 1965, p. 304–312 , doi : 10.1063 / 1.1704281 ( limited preview at SpringerLink [accessed on December 22, 2014]).

[Luttinger-3] JM Luttinger : An exactly soluable model of a many-fermion system . In: J. Math. Phys . tape 4 , no. 9 , 1963, pp. 1154-1162 , doi : 10.1063 / 1.1704046 .

[Haldane-4] FDM Haldane : 'Luttinger liquid theory' of one-dimensional quantum fluids . I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas. In: J. Phys. C . tape 14 , no. 19 , 1981, p. 2585-2919 , doi : 10.1088 / 0022-3719 / 14/19/010 .