The Lieb–Liniger model describes a gas of particles moving in one dimension and satisfying Bose–Einstein statistics.
A model of a gas of particles moving in one dimension and satisfying Bose–Einstein statistics was introduced in 1963  in order to study whether the available approximate theories of such gases, specifically Bogoliubov'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 Schrödinger equation.
The ground state as well as the low-lying excited states were computed and found to be in agreement with Bogoliubov'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 Bogoliubov'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 21st century, it became possible to produce this kind of gas using real atoms as particles.
Definition and solution of the modelEdit
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 function, 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 ansatz. 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
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.
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 , 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 Bogoliubov'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 Bogoliubov approximation (dashed) becomes worse as increases.
From three to one dimension.Edit
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 .
- Elliott H. Lieb and Werner Liniger, Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State, Physical Review 130: 1605–1616, 1963
- Elliott H. Lieb, Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum, Physical Review 130:1616–1624,1963
- Girardeau, Marvin (1960). "Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension". Journal of Mathematical Physics. 1 (6): 516–523. Bibcode:1960JMP.....1..516G. doi:10.1063/1.1703687.
- Dorlas, Teunis C. (1993). "Orthogonality and Completeness of the Bethe Ansatz Eigenstates of the nonlinear Schrödinger model". Communications in Mathematical Physics. 154 (2): 347–376. Bibcode:1993CMaPh.154..347D. doi:10.1007/BF02097001. S2CID 122730941.
- Lieb, Elliott H.; Seiringer, Robert; Yngvason, Jakob (2003). "One-dimensional Bosons in Three-dimensional Traps". Physical Review Letters. 91 (15): 150401. arXiv:cond-mat/0304071. Bibcode:2003PhRvL..91o0401L. doi:10.1103/PhysRevLett.91.150401. PMID 14611451. S2CID 5303148.
- Seiringer, Robert; Yin, Jun (2008). "The Lieb–Liniger Model as a Limit of Dilute Bosons in Three Dimensions". Communications in Mathematical Physics. 284 (2): 459–479. arXiv:0709.4022. Bibcode:2008CMaPh.284..459S. doi:10.1007/s00220-008-0521-6. S2CID 115173378.
- See also Elliott H. Lieb (2008), Scholarpedia, 3(12):8712.