The Bethe Ansatz

Another solution to the Yang-Baxter equation can be found, which has the same form as the rational \(R\)-matrix but slightly different matrix elements:

\begin{equation} R (\lambda) = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & b (\lambda) & c (\lambda) & 0 \\ 0 & c (\lambda) & b (\lambda) & 0 \\ 0 & 0 & 0 & 1 \end{array} \right), \hspace{1cm} b (\lambda) = \frac{\sinh \lambda}{\sinh (\lambda + \eta)}, \hspace{1cm} c (\lambda) = \frac{\sinh \eta}{\sinh (\lambda + \eta)}. \tag{a.r}\label{a.r} \end{equation}

The logic of the Algebraic Bethe Ansatz can again be applied here, following the previous example of the \(XXX\) case. Adopting the same convention for the reference state (choosing also the homogeneous limit with \(\xi = \eta/2\), but leaving \(\eta\) here as a free parameter) yields

\begin{equation} a (\lambda) = 1, \hspace{1cm} d(\lambda) = b(\lambda - \eta/2)^N = \left[ \frac{\sinh (\lambda - \eta/2)}{ \sinh (\lambda + \eta/2)} \right]^N. \tag{a.ad}\label{a.ad} \end{equation}

The Bethe equations read

\begin{equation*} \left[\frac{\sinh(\lambda_j + \eta/2)}{\sinh (\lambda_j - \eta/2)} \right]^N = \prod_{k (\neq j) = 1}^M \frac{\sinh (\lambda_j - \lambda_k + \eta)}{\sinh(\lambda_j - \lambda_k - \eta)}. \end{equation*}

The eigenvalue of the transfer matrix becomes

\begin{equation} \tau (\lambda | \{ \lambda_j \}) = \prod_{j=1}^M \frac{\sinh(\lambda_j - \lambda + \eta)}{\sinh(\lambda_j - \lambda)} + \left[\frac{\sinh(\lambda - \eta/2)}{\sinh(\lambda + \eta/2)}\right]^N \prod_{j=1}^M \frac{\sinh(\lambda_j - \lambda - \eta)}{\sinh(\lambda_j - \lambda)}. \tag{a.te}\label{a.te} \end{equation}

The first few conserved charges can be computed as

\begin{equation} P = -i \ln \tau (\lambda)|_{\lambda = \eta/2} \tag{a.q1}\label{a.q1} \end{equation}

giving eigenvalue

\begin{equation} P (\{ \lambda_j \}_M) = -i \sum_{j=1}^M \ln \frac{\sinh(\lambda_j + \eta/2)}{\sinh(\lambda_j - \eta/2)} . % = \pi M - \sum_{j=1}^M 2~\mbox{atan}~ \frac{\tanh \lambda_j}{\tanh \eta/2}. \tag{a.q1e}\label{a.q1e} \end{equation}

The Hamiltonian is (under an appropriate choice of the constant prefactor)

\begin{equation} H_{XXZ} = \frac{\sinh \eta}{2} \frac{d}{d\lambda} \ln \tau (\lambda) |_{\lambda = \eta/2} = \sum_{j=1}^N \left[S^x_j S^x_{j+1} + S^y_j S^y_{j+1} + \Delta (S^z_j S^z_{j+1} - \frac{1}{4}) \right] \tag{a.q2}\label{a.q2} \end{equation}

in which \(\Delta = \cosh \eta\). The energy eigenvalue of a state is

\begin{equation} E (\{ \lambda_j \}_M) = \sum_{j=1}^M \frac{\sinh^2 \eta}{\cosh 2\lambda_j - \cosh \eta}. \tag{a.q2e}\label{a.q2e} \end{equation}

For the gapped regime \(\Delta > 1\) we have that \(\eta\) is a real parameter. For the gapless regime \(0 < \Delta < 1\) it is convenient to reparametrize the anisotropy as \(\eta= -i\zeta\), so

\begin{equation} \Delta = \left\{ \begin{array}{cc} \cosh \eta, & 1 \leq \Delta, \nonumber \\ \cos \zeta, & -1 \leq \Delta \leq 1 \end{array} \right. \tag{da}\label{da} \end{equation}