The Bethe Ansatz

The lowest energy state will be obtained by forming a bound state of all \(N\) particles centered on zero 1964.McGuire.JMP.5, namely by choosing

\begin{equation*} \lambda^{N,a} = i \frac{\bar{c}}{2} (N+1 - 2a) + \mbox{O}(\delta)\,. \end{equation*}

The corresponding energy is

\begin{equation} E_{GS} = \sum_a (\lambda^{N,a})^2 = -\frac{\bar{c}^2}{4} \sum_{a=1}^N (N+1 - 2a)^2 = -\frac{\bar{c}^2}{12} N(N^2 - 1). \tag{al.e0}\label{al.e0} \end{equation}

Note that this goes like \(N^3\), unlike the repulsive case where the energy is not extensive. Although our results are not limited to this case, we will often consider the limit of a large number of particles \(N \gg 1\), with weak interactions \(\bar{c}\) such that the parameter \(g = \bar{c} N\) remains finite. In this case, the ground state energy per particle also remains finite, \(E_{GS} = -{g^2}/{12}\). We will find some similarly simplified limiting values for the correlations functions. It is however important to note that this is not a conventional thermodynamic limit with finite energy density as in the repulsive case.