Luttinger liquid
Encyclopedia
A Tomonaga-Luttinger liquid, more often referred to as simply a Luttinger liquid, is a theoretical model describing interacting electron
s (or other fermion
s) in a one-dimensional conductor
(e.g. quantum wire
s such as carbon nanotube
s). Such a model is necessary as the commonly used Fermi liquid
model breaks down for one dimension.
The Tomonaga-Luttinger liquid was first proposed by Tomonaga
in 1950. The model showed that under certain constraints, second-order interactions between electrons could be modelled as bosonic interactions. In 1963, Luttinger reformulated the theory in terms of Bloch sound waves and showed that the constraints proposed by Tomonaga were unnecessary in order to treat the second-order perturbations as bosons. But his solution of the model was incorrect, the correct one was given by Mattis and Lieb
1965.
Among the hallmark features of a Luttinger liquid are the following:
The Luttinger model is thought to describe the universal low-frequency/long-wavelength behaviour of any one-dimensional system of interacting fermions (that has not undergone a phase transition into some other state).
Among the physical systems believed to be described by the Luttinger model are:
Attempts to demonstrate Luttinger-liquid-like behaviour in those systems are the subject of ongoing experimental research in condensed matter physics
.
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s (or other fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s) in a one-dimensional conductor
Electrical conductor
In physics and electrical engineering, a conductor is a material which contains movable electric charges. In metallic conductors such as copper or aluminum, the movable charged particles are electrons...
(e.g. quantum wire
Quantum wire
In condensed matter physics, a quantum wire is an electrically conducting wire, in which quantum effects are affecting transport properties. Due to the quantum confinement of conduction electrons in the transverse direction of the wire, their transverse energy is quantized into a series of...
s such as carbon nanotube
Carbon nanotube
Carbon nanotubes are allotropes of carbon with a cylindrical nanostructure. Nanotubes have been constructed with length-to-diameter ratio of up to 132,000,000:1, significantly larger than for any other material...
s). Such a model is necessary as the commonly used Fermi liquid
Fermi liquid
Fermi liquid theory is a theoretical model of interacting fermions that describes the normal state of most metals at sufficiently low temperatures. The interaction between the particles of the many-body system does not need to be small...
model breaks down for one dimension.
The Tomonaga-Luttinger liquid was first proposed by Tomonaga
Sin-Itiro Tomonaga
was a Japanese physicist, influential in the development of quantum electrodynamics, work for which he was jointly awarded the Nobel Prize in Physics in 1965 along with Richard Feynman and Julian Schwinger.-Biography:...
in 1950. The model showed that under certain constraints, second-order interactions between electrons could be modelled as bosonic interactions. In 1963, Luttinger reformulated the theory in terms of Bloch sound waves and showed that the constraints proposed by Tomonaga were unnecessary in order to treat the second-order perturbations as bosons. But his solution of the model was incorrect, the correct one was given by Mattis and Lieb
Elliott H. Lieb
Elliott H. Lieb is an eminent American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis....
1965.
Among the hallmark features of a Luttinger liquid are the following:
- The response of the chargeCharge densityThe linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...
(or particleParticle densityThe particle density or true density of a particulate solid or powder, is the density of the particles that make up the powder, in contrast to the bulk density, which measures the average density of a large volume of the powder in a specific medium .The particle density is a relatively well-defined...
) density to some external perturbation are waves ("plasmonPlasmonIn physics, a plasmon is a quantum of plasma oscillation. The plasmon is a quasiparticle resulting from the quantization of plasma oscillations just as photons and phonons are quantizations of light and mechanical vibrations, respectively...
s" - or charge density waves) propagating at a velocity that is determined by the strength of the interaction and the average density. For a non-interacting system, this wave velocity is equal to the Fermi velocity, while it is higher (lower) for repulsive (attractive) interactions among the fermions.
- Likewise, there are spin density waves (whose velocity, to lowest approximation, is equal to the unperturbed Fermi velocity). These propagate independently from the charge density waves. This fact is known as spin-charge separationSpin-charge separationIn condensed matter physics, spin–charge separation is an unusual behavior of electrons in some materials in which they 'split' into two independent particles, the spinon and the chargon...
.
- ChargeCharge (physics)In physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges are associated with conserved quantum numbers.-Formal definition:...
and spinSpin (physics)In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
waves are the elementary excitations of the Luttinger liquid, unlike the quasiparticleQuasiparticleIn physics, quasiparticles are emergent phenomena that occur when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in free space...
s of the Fermi liquid (which carry both spin and charge). The mathematical description becomes very simple in terms of these waves (solving the one-dimensional wave equationWave equationThe wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
), and most of the work consists in transforming back to obtain the properties of the particles themselves (or treating impurities and other situations where 'backscattering' is important). See bosonizationBosonizationIn theoretical physics, one often studies two-dimensional conformal field theory. It has many very special properties. One of them is the equivalence of fermionic elementary fields and bosonic elementary fields. Bosonization also appears in 2D theories which aren't conformal field theories.The...
for one technique used.
- Even at zero temperature, the particles' momentum distribution function does not display a sharp jump, in contrast to the Fermi liquid (where this jump indicates the Fermi surface).
- There is no 'quasiparticle peak' in the momentum-dependent spectral function (i.e. no peak whose width becomes much smaller than the excitation energy above the Fermi level, as is the case for the Fermi liquid). Instead, there is a power-law singularity, with a 'non-universal' exponent that depends on the interaction strength.
- Around impurities, there are the usual Friedel oscillations in the charge density, at a wavevector of
. However, in contrast to the Fermi liquid, their decay at large distances is governed by yet another interaction-dependent exponent.
- At small temperatures, the scattering off these Friedel oscillations becomes so efficient that the effective strength of the impurity is renormalized to infinity, 'pinching off' the quantum wire. More precisely, the conductance becomes zero as temperature and transport voltage go to zero (and rises like a power law in voltage and temperature, with an interaction-dependent exponent).
- Likewise, the tunneling rate into a Luttinger liquid is suppressed to zero at low voltages and temperatures, as a power lawPower lawA power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...
.
The Luttinger model is thought to describe the universal low-frequency/long-wavelength behaviour of any one-dimensional system of interacting fermions (that has not undergone a phase transition into some other state).
Among the physical systems believed to be described by the Luttinger model are:
- artificial 'quantum wireQuantum wireIn condensed matter physics, a quantum wire is an electrically conducting wire, in which quantum effects are affecting transport properties. Due to the quantum confinement of conduction electrons in the transverse direction of the wire, their transverse energy is quantized into a series of...
s' (one-dimensional strips of electrons) defined by applying gate voltages to a two-dimensional electron gas, or by other means (lithographyLithographyLithography is a method for printing using a stone or a metal plate with a completely smooth surface...
, AFMAtomic force microscopeAtomic force microscopy or scanning force microscopy is a very high-resolution type of scanning probe microscopy, with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the optical diffraction limit...
, etc.) - electrons in carbon nanotubeCarbon nanotubeCarbon nanotubes are allotropes of carbon with a cylindrical nanostructure. Nanotubes have been constructed with length-to-diameter ratio of up to 132,000,000:1, significantly larger than for any other material...
s - electrons moving along edge states in the fractional Quantum Hall EffectFractional quantum Hall effectThe fractional quantum Hall effect is a physical phenomenon in which the Hall conductance of 2D electrons shows precisely quantised plateaus at fractional values of e^2/h. It is a property of a collective state in which electrons bind magnetic flux lines to make new quasiparticles, and excitations...
- electrons hopping along one-dimensional chains of molecules (e.g. certain organic molecular crystals)
- fermionic atoms in quasi-one-dimensional atomic traps
- a 1D 'chain' of half-odd-integer spins described by the Heisenberg modelHeisenberg model (quantum)The Heisenberg model is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spin of the magnetic systems are treated quantum mechanically...
(the Luttinger liquid model also works for integer spins if they are in a large enough magnetic field)
Attempts to demonstrate Luttinger-liquid-like behaviour in those systems are the subject of ongoing experimental research in condensed matter physics
Condensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
.
External links
- Short introduction (Stuttgart University, Germany)
- List of books (FreeScience Library)