The Bethe Ansatz

Let's consider the Lieb-Liniger model,

\begin{equation*} H_{LL} = \int_0^L dx \left\{ \Psi^\dagger (x) (-\partial_x^2) \Psi(x) + c \Psi^\dagger(x) \Psi^\dagger(x) \Psi(x) \Psi(x) - \mu \Psi^\dagger(x) \Psi(x) \right\} \end{equation*}

with canonical equal-time commutation relations \(\left[ \Psi(x), \Psi^\dagger (x') \right] = \delta(x-x')\).

Notations used here: system length is \(L\), number of particles is \(N = \int_0^L dx \Psi^\dagger (x) \Psi(x)\) (previously, the integer \(N\) was the number of sites, which now gets replaced by the system length given by the continuous parameter \(L\); don't confuse these for each other).

We define the Fourier transforms as

\begin{equation*} \Psi (x) = \frac{1}{L} \sum_k e^{ikx} \Psi_k, \hspace{10mm} \Psi_k = \int_0^L dx e^{-ikx} \Psi (x) \end{equation*}

so \(\left[ \Psi_k, \Psi^\dagger_{k'} \right] = L \delta_{k k'}\) and the Hamiltonian is

\begin{equation*} H_{LL} = \frac{1}{L} \sum_k (k^2 - \mu) \Psi^\dagger_k \Psi_k + \frac{c}{L^3} \sum_{k_1 k_2 q} \Psi^\dagger_{k_1 + q} \Psi^{\dagger}_{k_2 - q} \Psi_{k_2} \Psi_{k_1}. \tag{HLLk}\label{HLLk} \end{equation*}