Lieb–Liniger model

Example

The Lieb–Liniger model describes $N$ identical bosons on a circle of length $L$ with a repulsive contact interaction of strength $c > 0$. In units with $\hbar = 2m = 1$, its Hamiltonian acts on symmetric wave functions $\psi(x_1, \ldots, x_N)$ with $x_j \in [0, L)$ periodic: $$H = -\sum_{j=1}^N \frac{\partial^2}{\partial x_j^2} + 2c \sum_{1 \le i < j \le N} \delta(x_i - x_j).$$ Elliott Lieb and Werner Liniger solved it in 1963 with the Bethe ansatz. In the region $x_1 < x_2 < \cdots < x_N$ the eigenfunctions are finite sums of plane waves $e^{i(k_{P1} x_1 + \cdots + k_{PN} x_N)}$ over permutations $P$, with real quasi-momenta $k_1, \ldots, k_N$. Periodicity forces the Bethe equations $$e^{i k_j L} = \prod_{l \ne j} \frac{k_j - k_l + i c}{k_j - k_l - i c}, \qquad j = 1, \ldots, N,$$ and the energy is $E = k_1^2 + \cdots + k_N^2$.

Check in two limits. For $c = 0$ every factor is $1$, so $e^{i k_j L} = 1$ and $k_j \in (2\pi/L)\Z$: these are free bosons on the circle. As $c \to \infty$ every factor tends to $-1$, so $e^{i k_j L} = (-1)^{N-1}$: the spectrum is that of free fermions, the Tonks–Girardeau gas.