In physics, the Lieb–Liniger model describes a gas of particles moving in one dimension and satisfying Bose–Einstein statistics. More specifically, it describes a one dimensional Bose gas with Dirac delta interactions. It is named after Elliott H. Lieb and Werner Liniger who introduced the model in 1963. The model was developed to compare and test Nikolay Bogolyubov's theory of a weakly interaction Bose gas.

Definition

Given N bosons moving in one-dimension on the x-axis defined from [0,L] 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 \delta is the Dirac delta function. The constant c denotes the strength of the interaction, c>0 represents a repulsive interaction and c<0 an attractive interaction. The hard core limit c\to\infty is known as the Tonks–Girardeau gas. For a collection of bosons, the wave function is unchanged under permutation of any two particles (permutation symmetry), i.e., for all i \neq j and \psi satisfies for all j. The delta function in the Hamiltonian gives rise to a boundary condition when two coordinates, say x_1 and x_2 are equal; this condition is that as, the derivative satisfies

Solution

The time-independent Schrödinger equation, is solved by explicit construction of \psi. Since \psi is symmetric it is completely determined by its values in the simplex \mathcal{R}, 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 N ! permutations, P, of the integers, and P maps 1,2,\dots,N to. The coefficients a(P), as well as the k's are determined by the condition, and this leads to a total energy with the amplitudes given by These equations determine \psi in terms of the k's. These lead to N equations: where are integers when N is odd and, when N is even, they take values. For the ground state the I's satisfy

Thermodynamic limit