The Bethe Ansatz

Single-particle excited states will be obtained by giving finite momentum to the ground state \(N\)-string,

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

Such states have energy above the ground state given by

\begin{equation*} \omega_{N} (\mu) \equiv E_\mu-E_{GS}=N \mu^2 = k_{\mu}^2/N \end{equation*}

where \(k_{\mu} = N \mu\) is the total state momentum. For these states, there is only one Bethe equation for the string center \(\mu\), namely \(\mu = 2\pi {I}/{NL}\) with \(I\) an integer, so that the momentum is quantized as for a free wave, \(k_{\mu} = 2\pi {I}/{L}\). In the limit of large \(N\), this energy band becomes flat and quasi-degenerate with the ground state.