A Luttinger liquid, or Tomonaga–Luttinger liquid, is a theoretical model describing interacting
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s (or other
fermion
In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s) in a one-dimensional
conductor (e.g.
quantum wire
In mesoscopic physics, a quantum wire is an electrically conducting wire in which quantum effects influence the transport properties. Usually such effects appear in the dimension of nanometers, so they are also referred to as nanowires.
Quantum ...
s such as
carbon nanotube
A carbon nanotube (CNT) is a tube made of carbon with a diameter in the nanometre range ( nanoscale). They are one of the allotropes of carbon. Two broad classes of carbon nanotubes are recognized:
* ''Single-walled carbon nanotubes'' (''S ...
s). Such a model is necessary as the commonly used
Fermi liquid
Fermi liquid theory (also known as Landau's Fermi-liquid theory) is a theoretical model of interacting fermions that describes the normal state of the conduction electrons in most metals at sufficiently low temperatures. The theory describes the ...
model breaks down for one dimension.
The Tomonaga–Luttinger's liquid was first proposed by
Sin-Itiro Tomonaga in 1950. The model showed that under certain constraints, second-order interactions between electrons could be modelled as bosonic interactions. In 1963,
J.M. 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 solution was given by and
Elliot H. Lieb 1965.
Theory
Luttinger liquid theory describes low energy excitations in a 1D electron gas as bosons. Starting with the free electron Hamiltonian:
is separated into left and right moving electrons and undergoes linearization with the approximation
over the range
:
Expressions for bosons in terms of fermions are used to represent the Hamiltonian as a product of two boson operators in a
Bogoliubov transformation.
The completed
bosonization can then be used to predict spin-charge separation. Electron-electron interactions can be treated to calculate correlation functions.
Features
Among the hallmark features of a Luttinger liquid are the following:
* The response of the
charge
Charge or charged may refer to:
Arts, entertainment, and media Films
* ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* '' Charge!!'', an album by The Aqu ...
(or
particle
In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from s ...
) density to some external perturbation are waves ("
plasmon
In physics, a plasmon is a quantum of plasma oscillation. Just as light (an optical oscillation) consists of photons, the plasma oscillation consists of plasmons. The plasmon can be considered as a quasiparticle since it arises from the quant ...
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 separation.
*
Charge
Charge or charged may refer to:
Arts, entertainment, and media Films
* ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* '' Charge!!'', an album by The Aqu ...
and
spin
Spin or spinning most often refers to:
* Spin (physics) or particle spin, a fundamental property of elementary particles
* Spin quantum number, a number which defines the value of a particle's spin
* Spinning (textiles), the creation of yarn or thr ...
waves are the elementary excitations of the Luttinger liquid, unlike the
quasiparticle
In condensed matter physics, a quasiparticle is a concept used to describe a collective behavior of a group of particles that can be treated as if they were a single particle. Formally, quasiparticles and collective excitations are closely relate ...
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 equation
The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light ...
), 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
bosonization 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 oscillation
In solid-state physics, Friedel oscillations,
named after French physicist Jacques Friedel, arise from localized perturbations in a metallic or semiconductor system caused by a defect in the Fermi gas or Fermi liquid. Friedel oscillations are a ...
s in the charge density, at a
wavevector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
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 of 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 law
In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a relative change in the other quantity proportional to the ...
.
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).
Physical systems
Attempts to demonstrate Luttinger-liquid-like behaviour in those systems are the subject of ongoing experimental research in
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
.
Among the physical systems believed to be described by the Luttinger model are:
* artificial '
quantum wire
In mesoscopic physics, a quantum wire is an electrically conducting wire in which quantum effects influence the transport properties. Usually such effects appear in the dimension of nanometers, so they are also referred to as nanowires.
Quantum ...
s' (one-dimensional strips of electrons) defined by applying gate voltages to a
two-dimensional electron gas
A two-dimensional electron gas (2DEG) is a scientific model in solid-state physics. It is an Fermi gas, electron gas that is free to move in two dimensions, but tightly confined in the third. This tight confinement leads to quantized energy levels ...
, or by other means (
lithography
Lithography () is a planographic method of printing originally based on the miscibility, immiscibility of oil and water. The printing is from a stone (lithographic limestone) or a metal plate with a smooth surface. It was invented in 1796 by ...
,
AFM, etc.)
* electrons in
carbon nanotube
A carbon nanotube (CNT) is a tube made of carbon with a diameter in the nanometre range ( nanoscale). They are one of the allotropes of carbon. Two broad classes of carbon nanotubes are recognized:
* ''Single-walled carbon nanotubes'' (''S ...
s
* electrons moving along edge states in the
fractional quantum Hall effect
The fractional quantum Hall effect (fractional QHE or FQHE) is the observation of precisely quantized plateaus in the Hall conductance of 2-dimensional (2D) electrons at fractional values of e^2/h, where ''e'' is the electron charge and ''h'' i ...
or integer
quantum Hall effect
The quantum Hall effect (or integer quantum Hall effect) is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance exhi ...
although the latter is often considered a more trivial example.
* 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 model (the Luttinger liquid model also works for integer spins in a large enough magnetic field)
* electrons in
lithium molybdenum purple bronze.
See also
*
Fermi liquid
Fermi liquid theory (also known as Landau's Fermi-liquid theory) is a theoretical model of interacting fermions that describes the normal state of the conduction electrons in most metals at sufficiently low temperatures. The theory describes the ...
Bibliography
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References
External links
Short introduction(Stuttgart University, Germany)
List of books(FreeScience Library)
{{Authority control
Theoretical physics
Statistical mechanics
Condensed matter physics