The Bethe Ansatz

In the repulsive case, given a proper set of quantum numbers \(\{ I \}\), the solution of the Bethe equations for the set of rapidities \(\{ \lambda \}\) exists and is unique 1969.Yang.JMP.10 due to the convexity of the Yang-Yang action. Furthermore all these solutions have real rapidities \(\lambda_i\). For the attractive case, the situation is completely different. We will define \(\bar{c}=-c > 0\) as the interaction parameter; let us rewrite our Bethe equations as

\begin{equation} e^{i \lambda_{\alpha} L} = \prod_{\beta \neq \alpha} \frac{\lambda_{\alpha} - \lambda_{\beta} - i\bar{c}} {\lambda_{\alpha} - \lambda_{\beta} + i\bar{c}}, \hspace{10mm} \alpha = 1, ..., N. \tag{al.be}\label{al.be} \end{equation}

Consider now a complex rapidity \(\lambda_{\alpha} = \lambda + i \eta\). The Bethe equation for this rapidity is

\begin{equation*} e^{i\lambda_{\alpha} L} = e^{i \lambda L - \eta L} = \prod_{\beta \neq \alpha} \frac{\lambda_{\alpha} - \lambda_{\beta} - i\bar{c}} {\lambda_{\alpha} - \lambda_{\beta} + i\bar{c}}. \end{equation*}

We consider finite \(N\) and \(L \rightarrow \infty\). If \(\eta > 0\), we have \(e^{-\eta L} \rightarrow 0\) on the left-hand side. Looking at the finite product on the right-hand side, we conclude that there must thus be a rapidity \(\lambda_{\alpha'}\) such that \(\lambda_{\alpha'} = \lambda_{\alpha} - i \bar{c} + \mbox{O}(e^{-\eta L})\). On the other hand, if \(\eta < 0\), we have \(e^{-\eta L} \rightarrow \infty\) on the left-hand side, and there must thus be a rapidity \(\lambda_{\alpha'}\) such that \(\lambda_{\alpha'} = \lambda_{\alpha} + i \bar{c} + \mbox{O}(e^{-|\eta| L})\). The rapidities thus like to arrange themselves into clusters, the elements of each cluster being evenly-spaced by \(\bar{c}\) in the imaginary direction. Such clusters represent bound states of particles, and we will call them strings.

For a given number of atoms \(N\), we can construct eigenstates with fixed string content by partitioning \(N\) into \(N_j\) strings of length \(j\), denoting the total number of strings as \(N_s\). We clearly have

\begin{equation*} N = \sum_{j} j N_j, \hspace{1cm} N_s = \sum_{j} N_j. \end{equation*}

Specifically, we will parametrize the string rapidities as

\begin{equation} \lambda_{\alpha}^{j, a} = \lambda_{\alpha}^j + i\frac{\bar{c}}{2} ( j + 1 - 2a) + i\delta_{\alpha}^{j,a}, \tag{al.sh}\label{al.sh} \end{equation}

with exponentially small deviations \(\delta \sim e^{-(cst)L}\) provided \(N_s/L \rightarrow 0\). In our string notation, the index \(a = 1, \dots, j\) labels rapidities within the string, and \(\alpha = 1, \dots, N_j\) labels strings of a given length.

We stress that perfect strings (i.e. with all the \(\delta_i=0\)) are exact eigenstates in the limit \(L\to\infty\) with \(N_s/L \rightarrow 0\) for arbitrary \(N\). It is then natural to consider the limit \(L\to\infty\) at fixed \(N\). This is different from what done in the repulsive case where the limit \(N,L\to\infty\) at fixed density \(N/L\) is performed. Here, the \(N\) particles remain strongly correlated and bound to one another even when \(L \to \infty\).

These bound states should be viewed as individual particles of mass \(j\), with momentum and energy of the string centered on \(\lambda^j_{\alpha}\) given by

\begin{equation} E_{(j, \alpha)} = j (\lambda^j_{\alpha})^2 - \frac{\bar{c}^2}{12} j(j^2 - 1), \hspace{1cm} P_{(j,\alpha)} = j \lambda^j_{\alpha}. \tag{al.eps}\label{al.eps} \end{equation}

Such strings are known but not commonly discussed in the literature on the Bose gas, since they do not appear in the repulsive case. However, their direct equivalents exist in integrable spin chains, where they have been extensively studied. The technology to treat them, at least on the level of eigenstates, is thus completely standard.