Lieb–Liniger model

In physics, the Lieb–Liniger model describes a gas of particles moving in one dimension and satisfying Bose–Einstein statistics. It is sometimes called a one dimensional Bose gas with Dirac delta interaction. It also can be considered as quantum non-linear Schrödinger equation. It is named after Elliott H. Lieb and Werner Liniger who introduced the model in 1963.

Definition

Fig. 1: The ground state energy, from. See text.

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

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

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.

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 $\left.\left({\frac {\partial }{\partial x_{2}}}-{\frac {\partial }{\partial x_{1}}}\right)\psi (x_{1},x_{2})\right|_{x_{2}=x_{1}+}=c\psi (x_{1}=x_{2})$. The hard core limit $c\to \infty$ is known as the Tonks–Girardeau gas.

Solution

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 as 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

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