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.

In the hard interaction limit, this model recovers the Tonks–Girardeau model.

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 $c\geq 0$ denotes the repulsive strength of the interaction. The delta function 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 $c=\infty$ 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 $j$.

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 $0\leq x_{1}\leq x_{2}\leq \dots ,\leq x_{N}\leq L$.

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

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

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

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

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

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

External links

• See also Elliott H. Lieb (2008), Scholarpedia, 3(12):8712.[1]