Lieb–Liniger model

The Lieb–Liniger model describes a gas of particles moving in one dimension and satisfying Bose–Einstein statistics.

IntroductionEdit

A model of a gas of particles moving in one dimension and satisfying Bose–Einstein statistics was introduced in 1963 [1][2] 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.[3]

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

 
 

Dorlas (1993) proved that all eigenfunctions of   are of this form.[4]

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.

Thermodynamic limitEdit

 
Fig. 1: The ground state energy, from.[1] See text.

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  ,  .

 
Fig. 2: The energies of the two types of excitations, from.[2] See text.

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[5] and for excited states.[6] 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  .

ReferencesEdit

  1. ^ a b 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
  2. ^ a b Elliott H. Lieb, Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum, Physical Review 130:1616–1624,1963
  3. ^ 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.
  4. ^ 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.
  5. ^ 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.
  6. ^ 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.

External linksEdit

  • See also Elliott H. Lieb (2008), Scholarpedia, 3(12):8712.[1]