The Bethe Ansatz

The Bethe equations for the \(XXX\) magnet allow solutions which are not restricted to the real line. Complex rapidities can occur. These complex rapidities organize themselves into self-conjugate patterns called {\it strings}, rapidities within an \(n\)-string being reparametrized as

\begin{equation} \lambda_\alpha^{n, a} = \lambda_\alpha^n + \frac{i}{2} (n+1 - 2a) + i \delta_\alpha^{n,a}, \hspace{10mm} a = 1, ..., n. \tag{h.ls}\label{h.ls} \end{equation}

The real parameter \(\lambda_\alpha^j\) represents the {\it string center}, namely the `center of mass' of the composite object represented by the \(n\) rapidities. The string deviations \(\delta_\alpha^{j,a}\) are in most circumstances exponentially small in system size, and can typically be neglected (there are important exceptions to this rule). Since complex rapidities represent bound states of downturned spins (as can be seen from the Bethe Ansatz wavefunction itself), a cluster of the form h.ls represents a single bona fide particle. Out of \(n\) rapidities within a string, we thus get only one independent parameter, the string center \(\lambda_\alpha^{j}\).

The string hypothesis assumes that eigenstates of the Heisenberg chain are represented by sets of rapidities which organize themselves into (perfect) strings. An eigenstate with \(M\) down spins is then understood as an eigenstate with \(M_j\) strings of length \(j\), the total number of downturned spins obeying

\begin{equation*} \sum_{j=1}^\infty j M_j = M. \end{equation*}

The total number of strings in an eigenstate is given by

\begin{equation*} \sum_{j=1}^\infty M_j \equiv N_s. \end{equation*}

In a given eigenstate, the number of independent parameters is thus \(N_s\) (the number of string centers) instead of \(M\). The Bethe equations for strings can be rewritten by taking the product of h.be over all rapidities within a string. Doing this, one finds (for \(N\) even) the reduced set of Bethe equations

\begin{equation} \bar{e}_j^N (\lambda_{\alpha}^n) = (-1)^{M_j-1}\prod_{(k,\beta) \neq (j,\alpha)} \bar{E}_{jk} (\lambda_{\alpha}^j - \lambda_{\beta}^k), \tag{h.bgt}\label{h.bgt} \end{equation}

where

\begin{equation} \bar{e}_j(\lambda) = -\frac{\lambda + j \frac{i}{2}}{\lambda - j \frac{i}{2}}, \hspace{1cm} \bar{E}_{jk}(\lambda) = \bar{e}_{|j-k|}^{1-\delta_{j k}} (\lambda) \bar{e}_{|j-k|+2}^2(\lambda) ... \bar{e}_{j+k-2}^2 (\lambda) \bar{e}_{j+k}(\lambda). \tag{h.eE}\label{h.eE} \end{equation}

Taking logs, we obtain the following reduced set of equations (which we shall call the Bethe-Gaudin-Takahashi equations)

\begin{equation} \phi_j (\lambda^j_\alpha) - \frac{1}{N} \sum_{k=1}^{N_s} \sum_{\beta = 1}^{M_k} \Phi_{jk} (\lambda^j_\alpha - \lambda^k_\beta) = \frac{2\pi}{N} I^j_\alpha \tag{h.bgtl}\label{h.bgtl} \end{equation}

in which

\begin{equation*} I_\alpha^j \in \left\{ \begin{array}{cc} \mathbb{Z} + \frac{1}{2}, & M_j ~\mbox{even} \\ \mathbb{Z}, & M_j ~\mbox{odd}. \end{array} \right. \end{equation*}

The kernels \(\phi_j\) are defined as

\begin{equation*} \phi_j(\lambda) = 2\mbox{atan} \frac{2\lambda}{n} \end{equation*}

and the string-string scattering phase shift is

\begin{equation} \Phi_{jk} (\lambda) = (1 - \delta_{jk}) \phi_{|j-k|} (\lambda) + 2\phi_{|j-k|+2} (\lambda) + ... + 2\phi_{j+k-2} (\lambda) + \phi_{j+k} (\lambda). \tag{h.phijk}\label{h.phijk} \end{equation}

These have the simple limits

\begin{equation*} \lim_{\lambda \rightarrow \infty} \phi_j (\lambda) = \pi, \hspace{10mm} \lim_{\lambda \rightarrow \infty} \Phi_{jk} (\lambda) = (2\min(j,k) - \delta_{jk}) \pi. \end{equation*}