The Bethe Ansatz

These are obtained by splitting up the ground-state \(N\) string in two pieces. In general, consider having an \(N-M\) and an \(M\) string:

\begin{align*} \mu^{N-M,a} &= \mu_s + i \frac{\bar{c}}{2} (N - M +1 - 2a) + \mbox{O}(\delta), \hspace{1cm} a = 1, ..., N-M, \nonumber \\ \mu^{M,a} &= \mu_M + i \frac{\bar{c}}{2} (M + 1 - 2a) + \mbox{O}(\delta), \hspace{1cm} a = 1, ..., M. \end{align*}

The energy of this state above the ground state is given by

\begin{equation*} \omega_{N-M:M} (\mu_s, \mu_M) = \omega^0_{N-M:M} + (N-M) \mu_s^2 + M \mu_M^2, \end{equation*}

where we have defined the rest energy

\begin{equation*} \omega^0_{N-M:M} = \frac{\bar{c}^2}{4} N M (N-M). \end{equation*}

The total momentum is the sum of the two string momenta,

\begin{equation*} k = k_s + k_M = (N-M) \mu_s + M \mu_M, \end{equation*}

so we can write the energy as

\begin{equation*} \omega_{N-M:M} (k_s, k_M) = \omega^0_{N-M:M} + \frac{k_s^2}{N-M} + \frac{k_M^2}{M}. \end{equation*}

Similarly to the single-particle case, the Bethe equations are here very simple, namely

\begin{align*} (N-M) \mu_s L - \Phi_{N-M,M} (\mu_s - \mu_M) &= 2\pi I_s, \nonumber \\ M \mu_M L + \Phi_{N-M,M} (\mu_s - \mu_M) &= 2\pi I_M, \end{align*}

with \(I_s, I_M\) integers. In the limit of large L, we can thus ignore the scattering phase shift, and take \(\mu_s\) and \(\mu_M\) as free parameters. The total momentum \(k\) of the state can take on any value \(2\pi I/L\), but the energy is bounded from below by

\begin{equation*} \omega^l_{N-M:M} (k) = \omega^0_{N-M:M} + \frac{k^2}{N}. \end{equation*}

Given external frequency \(\omega\) and momentum \(k\) parameters, there are two solutions to the dynamical constraints, namely

\begin{align*} \mu_s^{\pm}(k,\omega) &= \frac{k}{N} \mp \left[\frac{M}{N(N-M)}\right]^{1/2} [\omega - \omega^l_{N-M:M}(k)]^{1/2}, \nonumber \\ \mu_M^{\pm}(k,\omega) &= \frac{k}{N} \pm \left[\frac{N-M}{NM}\right]^{1/2} [\omega - \omega^l_{N-M:M}(k)]^{1/2}. \end{align*}

Therefore, in the large \(L\) limit, these states for a two-fold degenerate continuum beginning at the lower threshold \(\omega^l_{N-M:M} (k)\) and extending to arbitrarily high energy,

\begin{equation*} \mbox{N-M:M continuum:} \hspace{1cm} \omega^l_{N-M:M}(k) \leq \omega < \infty. \label{eq:1DBG:N-M:Mcontinuum} \end{equation*}

For finite \(L\), this is of course not strictly a continuum: only discrete energy levels \(\omega\) then exist, as determined from the Bethe equations.