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. t 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
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


 * $$ \psi(x_1, \dots, x_N) = \sum_P a(P)\exp \left( i \sum_{j=1}^N k_{P j} x_j\right)  $$,

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 $$\pm \frac12, \pm \frac32, \dots$$. 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. $$