Lieb-Liniger Model
Encyclopedia
The Lieb–Liniger model describes a gas of particles moving in one dimension and satisfying Bose–Einstein statistics
.
Bose–Einstein statistics
was introduced in 1963 in order to study whether the available approximate theories
of such gases, specifically Bogolubov's theory, would conform to the
actual properties of the model gas. The model is based on a well
defined
Schrödinger Hamiltonian for particles interacting with each other
via a two-body potential, and all the eigenfunctions and eigenvalues of
this Hamiltonian can, in principle, be calculated exactly. Sometimes it is called one dimensional Bose gas
with delta interaction. It also can be considered as quantum Non-linear Schroedinger equation.
The ground state as well as the low-lying excited states were computed
and found to be in agreement with Bogolubov's theory when the potential
is small, except for the fact that there are actually two types of
elementary excitations instead of one, as predicted by Bogolubov's and
other theories.
The model seemed to be only of academic interest until, with the
sophisticated experimental techniques developed in the first decade of
the 21
century, it became possible to produce this
kind of gas using real atoms as particles.
particles with coordinates 
on the line 
, with periodic boundary conditions. Thus, an allowed wave function 
is symmetric, i.e., 
for all 
and 
satisfies 
for all 
. The Hamiltonian, in appropriate units, is
Bose–Einstein statistics
In statistical mechanics, Bose–Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.-Concept:...
.
Introduction
A model of a gas of particles moving in one dimension and satisfyingBose–Einstein statistics
Bose–Einstein statistics
In statistical mechanics, Bose–Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.-Concept:...
was introduced in 1963 in order to study whether the available approximate theories
of such gases, specifically Bogolubov's theory, would conform to the
actual properties of the model gas. The model is based on a well
defined
Schrödinger Hamiltonian for particles interacting with each other
via a two-body potential, and all the eigenfunctions and eigenvalues of
this Hamiltonian can, in principle, be calculated exactly. Sometimes it is called one dimensional Bose gas
Bose gas
An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose–Einstein statistics...
with delta interaction. It also can be considered as quantum Non-linear Schroedinger equation.
The ground state as well as the low-lying excited states were computed
and found to be in agreement with Bogolubov's theory when the potential
is small, except for the fact that there are actually two types of
elementary excitations instead of one, as predicted by Bogolubov's and
other theories.
The model seemed to be only of academic interest until, with the
sophisticated experimental techniques developed in the first decade of
the 21
century, it became possible to produce thiskind of gas using real atoms as particles.
Definition and solution of the Model
There are particles with coordinates on the line , with periodic boundary conditions. Thus, an allowed wave function is symmetric, i.e., for all and satisfies for all . The Hamiltonian, in appropriate units, is-
where
is the Dirac delta functionDirac delta functionThe Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
, i.e., the interaction is a contact interaction. The constant
denotes its strength. The delta function gives rise to a boundary condition when two coordinates, say 
and 
are equal; this condition is that as 
, the derivative satisfies 
. The hard core limit 
is known as the Tonks–Girardeau gas.
Schrödinger's time independent equation,
is solved by explicit construction of 
. Since 
is symmetric it is completely determined by its values in the simplex 
, defined by the condition that 
. In this region one looks for a 
of the form considered by H.A. Bethe in 1931 in the context of magnetic spin systems—the Bethe ansatzBethe ansatzIn physics, the Bethe ansatz is a method for finding the exact solutions of certain one-dimensional quantum many-body models. It was invented by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic Heisenberg model Hamiltonian...
. That is, for certain real numbers
, to be determined,
-
where the sum is over all
permutations, 
, of the integers 
, and 
maps 
to 
. The coefficients 
, as well as the 
's are determined by the condition 
, and this leads to
-
Dorlas (1993) proved that all eigenfunctions of
are of this form.
These equations determine
in terms of the 
's, which, in turn, are determined by the periodic boundary conditions. These lead to 
equations:
-
where
are integers when 
is odd and, when 
is even, they take values 
. For the ground state the 
's satisfy
-
The first kind of elementary excitation consists in choosing
as before, but increasing 
by an amount 
(or decreasing 
by 
). The momentum of this state is 
(or 
).
For the second kind, choose some
and increase 
for all 
. The momentum of this state is 
. Similarly, there is a state with 
. The momentum of this type of excitation is limited to 
These excitations can be combined and repeated many times. Thus, they are bosonic-like. If we denote the ground state (= lowest) energy by
and the energies of the states mentioned above by 
then 
and 
are the excitation energies of the two modes. Later the model was also solved by Quantum inverse scattering methodQuantum inverse scattering methodQuantum inverse scattering method relates two different approaches:1) Inverse scattering transform is a method of solving classical integrable differential equations of evolutionary type.Important concept is Lax representation....
. This help to calculate correlation functions and to find conservation laws. The Hamiltonian of the model has a family of commuting operators: conservation laws. An explicit expression for higher conservation laws can be found in Davies and Korepin (1989).
Thermodynamic limit
To discuss a gas we take a limit
and 
to
infinity with the density
fixed. The ground state
energy per particle
, and the
all have limits as 
. While there are two parameters, 
and
, simple length scaling 
shows
that there is really only one, namely
.
To evaluate
we assume that the N 
's lie between numbers 
and −K, to be determined, and with a density 
. This 
is found to satisfy the equation (in the interval 
)
-
which has a unique positive solution. An excitation distorts this density
and similar integral equations determine these distortions. The ground state energy per particle is given by
Figure 1 shows how
depends on 
and also shows Bogolubov's approximation to
. The latter is asymptotically exact to second order in 
, namely, 
. At 
, 
.
Figure 2 shows the two excitation energies
and 
for a small value of 
. The two curves are similar to these for all values of 
, but the Bogolubov approximation (dashed) becomes worse as 
increases.
Correlation function (statistical mechanics)Correlation function (statistical mechanics)In statistical mechanics, the correlation function is a measure of the order in a system, as characterized by a mathematical correlation function, and describes how microscopic variables at different positions are correlated....
of the model was evaluated in the book entitled of Quantum Inverse Scattering Method and Correlation Functions by
V.E. Korepin, N.M. Bogoliubov and A.G. Izergin (Cambridge University Press, 1993).
Quantum entanglementQuantum entanglementQuantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
of the model was calculated in Physical Review Letters, vol 92, issue 9, electronic identifier 096402, 05 March 2004 .
From three to one dimension.
This one-dimensional gas can be made using real, three-dimensional atoms as particles. One can prove, mathematically, from the Schrödinger equation for three-dimensional particles in a long cylindrical container, that the low energy states are described by the one-dimensional Lieb–Liniger model. This was done for the ground state and for excited states. The cylinder does not have to be as narrow as the atomic diameter; it can be much wider if the excitation energy in the direction perpendicular to the axis is large compared to the energy per particle
.
External links
- See also Elliott H. Lieb (2008), Scholarpedia, 3(12):8712.http://www.scholarpedia.org/article/Lieb-Liniger_model_of_a_Bose_Gas
- See also V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, 1993.http://books.google.com/books?id=kaZ0pKIHhxAC&dq=quantum+inverse+scattering+method&printsec=frontcover&source=bl&ots=4AaoICh4Q3&sig=8RoPelCcAOLTG1vY21DDLCulNGY&hl=en&ei=I6zuScSBE5a-M9q1pOwP&sa=X&oi=book_result&ct=result&resnum=8#v=onepage&q&f=false
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