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 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 $\psi (x_{1},x_{2},\dots ,x_{j},\dots ,x_{N})$. The Hamiltonian, of this model is introduced as

$H=-\sum _{i=1}^{N}{\frac {\partial ^{2}}{\partial x_{i}^{2}}}+2c\sum _{i=1}^{N}\sum _{j>i}^{N}\delta (x_{i}-x_{j})\ ,$

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., $\psi (\dots ,x_{i},\dots ,x_{j},\dots )=\psi (\dots ,x_{j},\dots ,x_{i},\dots )$ for all $i\neq j$ and $\psi$ satisfies $\psi (\dots ,x_{j}=0,\dots )=\psi (\dots ,x_{j}=L,\dots )$ 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 $x_{2}\searrow x_{1}$, the derivative satisfies

$\left.\left({\frac {\partial }{\partial x_{2}}}-{\frac {\partial }{\partial x_{1}}}\right)\psi (x_{1},x_{2})\right|_{x_{2}=x_{1}+}=c\psi (x_{1}=x_{2})$.

Solution

Fig. 1: The ground state energy (per particle) $e$ as a function of the interaction strength per density $\gamma =Lc/N$, from.

The time-independent Schrödinger equation $H\psi =E\psi$, 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 $0\leq x_{1}\leq x_{2}\leq \dots ,\leq x_{N}\leq L$.

The solution can be written in the form of a Bethe ansatz as

$\psi (x_{1},\dots ,x_{N})=\sum _{P}a(P)\exp \left(i\sum _{j=1}^{N}k_{Pj}x_{j}\right)$,

with wave vectors $0\leq k_{1}\leq k_{2}\leq \dots ,\leq k_{N}$, where the sum is over all $N!$ permutations, $P$, of the integers $1,2,\dots ,N$, and $P$ maps $1,2,\dots ,N$ to $P_{1},P_{2},\dots ,P_{N}$. The coefficients $a(P)$, as well as the $k$'s are determined by the condition $H\psi =E\psi$, and this leads to a total energy

$E=\sum _{j=1}^{N}\,k_{j}^{2}$,

with the amplitudes given by

$a(P)=\prod _{1\leq i<j\leq N}\left(1+{\frac {ic}{k_{Pi}-k_{Pj}}}\right)\,.$

These equations determine $\psi$ in terms of the $k$'s. These lead to $N$ equations:

$L\,k_{j}=2\pi I_{j}\ -2\sum _{i=1}^{N}\arctan \left({\frac {k_{j}-k_{i}}{c}}\right)\qquad \qquad {\text{for }}j=1,\,\dots ,\,N\ ,$

where $I_{1}<I_{2}<\cdots <I_{N}$ are integers when $N$ is odd and, when $N$ is even, they take values $\pm {\frac {1}{2}},\pm {\frac {3}{2}},\dots$ . For the ground state the $I$'s satisfy

$I_{j+1}-I_{j}=1,\quad {\rm {for}}\ 1\leq j<N\qquad {\text{and }}I_{1}=-I_{N}.$