The Bethe Ansatz

Excited states

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.

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Author: Jean-Sébastien Caux

Created: 2024-01-18 Thu 14:24