Lieb–Liniger model

Template:Short description Template:Use dmy datesIn 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 Template:Ill who introduced the model in 1963. The model was developed to compare and test Nikolay Bogolyubov's theory of a weakly interacting Bose gas. It can be seen as one model in the theory of generalized hydrodynamics.

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 $c\to \mathrm{\infty }$ 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 $i\ne j$ 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. The condition is that as $x_2$ approaches $x_1$ from above ($x_2\searrow x_1$), the derivative satisfies

${(\frac{\partial }{\partial x_2}-\frac{\partial }{\partial x_1})\psi (x_1,x_2)|}_{x_2=x_1+}=c\psi (x_1=x_2)$.

Solution

File:Lieb-liniger-1.gif

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 $ℛ$, defined by the condition that $0\le x_1\le x_2\le \dots ,\le x_N\le L$.

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 $0\le k_1\le k_2\le \dots ,\le k_N$, 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

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

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

$L{k}_j=2\pi I_j-2\sum _{i=1}^N\arctan \left(\frac{k_j-k_i}{c}\right)\text{for }j=1,\dots ,N,$

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

$I_{j+1}-I_j=1,\mathrm{f}\mathrm{o}\mathrm{r}1\le j<N\text{and }{I}_1=-I_N.$