Lieb–Liniger model

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.

There are ${\displaystyle N}$ particles with coordinates ${\displaystyle x}$ on the line ${\displaystyle [0,L]}$, with periodic boundary conditions. Thus, an allowed wave function ${\displaystyle \psi (x_{1},x_{2},\dots ,x_{j},\dots ,x_{N})}$ is symmetric, i.e., ${\displaystyle \psi (\dots ,x_{i},\dots ,x_{j},\dots )=\psi (\dots ,x_{j},\dots ,x_{i},\dots )}$ for all ${\displaystyle i\neq j}$ and ${\displaystyle \psi }$ satisfies ${\displaystyle \psi (\dots ,x_{j}=0,\dots )=\psi (\dots ,x_{j}=L,\dots )}$ for all ${\displaystyle j}$. The Hamiltonian, in appropriate units, is

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