# The Bethe Ansatz

##### Excited statesc.cm.al.e

Excitations above the ground state are then obtained by either simply giving momentum to the ground state $$N$$ string, or more elaborately by partitioning it into smaller strings to which individual momenta can be given. We consider here only the $$N$$ atom sector relevant for the dynamical structure factor. The case of states with $$N-1$$ (or less) atoms trivially follows.

We will label the string content of eigenstates by column-separated entries specifying the length and number of each different string type. For example $$N-M:M$$ will be a state with a $$N-M$$ string and a $$M$$ string, and $$N-M_1-2M_2:M_1:(M_2)_2$$ a state with an $$N - M_1 - 2M_2$$ string, an $$M_1$$ string and two $$M_2$$ strings.

###### Single-particle states

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.

###### Two-particle states

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. Except where otherwise noted, all content is licensed under a Creative Commons Attribution 4.0 International License.

Created: 2023-06-07 Wed 16:02