In physics, the LiebâÂÂLiniger model describes a gas of particles moving in one dimension and satisfying BoseâÂÂEinstein statistics. More specifically, it describes a one dimensional Bose gas with Dirac delta interactions. It is named after Elliott H. Lieb and who introduced the model in 1963. The model was developed to compare and test Nikolay Bogolyubov's theory of a weakly interacting Bose gas. It can be seen as one model in the theory of generalized hydrodynamics.
Given bosons moving in one-dimension on the -axis defined from with periodic boundary conditions, a state of the N-body system must be described by a many-body wave function . The Hamiltonian, of this model is introduced as
where is the Dirac delta function. The constant denotes the strength of the interaction, represents a repulsive interaction and an attractive interaction. The hard core limit is known as the TonksâÂÂGirardeau gas.
For a collection of bosons, the wave function is unchanged under permutation of any two particles (permutation symmetry), i.e., for all and satisfies for all .
The delta function in the Hamiltonian gives rise to a boundary condition when two coordinates, say and are equal. The condition is that as approaches from above (), the derivative satisfies
The time-independent Schrödinger 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 .
The solution can be written in the form of a Bethe ansatz as
with wave vectors , 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 a total energy
with the amplitudes given by
These equations determine in terms of the 's. These lead to equations:
where are integers when is odd and, when is even, they take values . For the ground state the 's satisfy