Lieb–Liniger model

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 [de] who introduced the model in 1963.

[1] The model was developed to compare and test Nikolay Bogolyubov's theory of a weakly interaction Bose gas.

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

ψ (

The Hamiltonian, of this model is introduced as where

is the Dirac delta function.

denotes the strength of the interaction,

represents a repulsive interaction and

an attractive interaction.

[3] The hard core limit

[3] For a collection of bosons, the wave function is unchanged under permutation of any two particles (permutation symmetry), i.e.,

ψ ( … ,

, … ) = ψ ( … ,

ψ

ψ ( … ,

= 0 , … ) = ψ ( … ,

The delta function in the Hamiltonian gives rise to a boundary condition when two coordinates, say

), the derivative satisfies The time-independent Schrödinger equation

ψ =

, is solved by explicit construction of

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[2] with wave vectors

's are determined by the condition

, and this leads to a total energy with the amplitudes given by These equations determine

For the ground state the